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Solving Linear Equations and Inequalities

23 Solve Linear Inequalities

Learning Objectives

Past the cease of this section, you will be able to:

Graph inequalities on the number line
Solve inequalities using the Subtraction and Improver Properties of inequality
Solve inequalities using the Division and Multiplication Properties of inequality
Solve inequalities that crave simplification
Translate to an inequality and solve

Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true argument when substituted into the equation.

What about the solution of an inequality? What number would make the inequality $x>3$ truthful? Are you thinking, 'x could be four'? That's correct, just 10 could be 5 likewise, or xx, or fifty-fifty 3.001. Whatsoever number greater than iii is a solution to the inequality $x>3$ .

We show the solutions to the inequality $x>3$ on the number line by shading in all the numbers to the right of iii, to show that all numbers greater than iii are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of $x>3$ is shown in (Figure). Delight annotation that the following convention is used: light blue arrows point in the positive direction and night blue arrows point in the negative direction.

The inequality $x>3$ is graphed on this number line.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than 3 is graphed on the number line, with an open parenthesis at x equals 3, and a red line extending to the right of the parenthesis.

The graph of the inequality $x\ge 3$ is very much like the graph of $x>3$ , simply now nosotros need to evidence that 3 is a solution, also. We do that by putting a subclass at $x=3$ , every bit shown in (Figure).

The inequality $x\ge 3$ is graphed on this number line.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is greater than or equal to 3 is graphed on the number line, with an open bracket at x equals 3, and a red line extending to the right of the bracket.

Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open up bracket symbol, [, shows that the endpoint is included.

Graph on the number line:

ⓐ $x\le 1$ ⓑ $x<5$ ⓒ $x>-1$

Nosotros tin can likewise represent inequalities using interval notation. As nosotros saw above, the inequality $x>3$ means all numbers greater than 3. At that place is no upper end to the solution to this inequality. In interval notation, nosotros express $x>3$ as $\left(3,\infty \right).$ The symbol $\infty$ is read equally 'infinity'. Information technology is non an actual number. (Figure) shows both the number line and the interval notation.

The inequality $x>3$ is graphed on this number line and written in interval notation.

The inequality $x\le 1$ means all numbers less than or equal to 1. At that place is no lower cease to those numbers. Nosotros write $x\le 1$ in interval notation as $\left(\text{−}\infty ,1\right]$ . The symbol $\text{−}\infty$ is read as 'negative infinity'. (Figure) shows both the number line and interval annotation.

The inequality $x\le 1$ is graphed on this number line and written in interval notation.

This figure is a number line ranging from negative 5 to 5 with tick marks for each integer. The inequality x is less than or equal to 1 is graphed on the number line, with an open bracket at x equals 1, and a red line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 1, bracket.

Inequalities, Number Lines, and Interval Notation

Did you notice how the parenthesis or subclass in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in (Figure).

The notation for inequalities on a number line and in interval note use similar symbols to express the endpoints of intervals.

This figure shows the same four number lines as above, with the same interval notation labels. Below the interval notation for each number line, there is text indicating how the notation on the number lines is similar to the interval notation. The first number line is a graph of x is greater than a, and the interval notation is parenthesis, a comma infinity, parenthesis. The text below reads:

Graph on the number line and write in interval notation.

ⓐ $x\ge -3$ ⓑ ⓒ $x\le -\frac{3}{5}$

Graph on the number line and write in interval note:

ⓐ $x>2$ ⓑ $x\le -1.5$ ⓒ $x\ge \frac{3}{4}$

Graph on the number line and write in interval notation:

ⓐ $x\le -4$ ⓑ $x\ge 0.5$ ⓒ $ten<-\frac{2}{3}$

Solve Inequalities using the Subtraction and Add-on Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results volition be equal.

Properties of Equality

$\begin{array}{cccc}\mathbf{\text{Subtraction Property of Equality}}\hfill & & & \mathbf{\text{Addition Property of Equality}}\hfill \\ \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill & & & \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill \\ \begin{array}{cccc}\text{if}\hfill & \hfill a& =\hfill & b,\hfill \\ \text{then}\hfill & \hfill a-c& =\hfill & b-c.\hfill \end{array}\hfill & & & \begin{array}{cccc}\text{if}\hfill & \hfill a& =\hfill & b,\hfill \\ \text{then}\hfill & \hfill a+c& =\hfill & b+c.\hfill \end{array}\hfill \end{array}$

Similar properties agree truthful for inequalities.

For example, we know that −4 is less than 2.
If we subtract 5 from both quantities, is the left side notwithstanding less than the right side?
We become −9 on the left and −3 on the right.
And we know −nine is less than −3.
	The inequality sign stayed the same.

Similarly we could bear witness that the inequality also stays the same for addition.

This leads u.s.a. to the Subtraction and Add-on Properties of Inequality.

Properties of Inequality

$\brainstorm{assortment}{cccc}\mathbf{\text{Subtraction Belongings of Inequality}}\hfill & & & \mathbf{\text{Addition Property of Inequality}}\hfill \\ \text{For whatever numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\dominion{0.2em}{0ex}}c,\hfill & & & \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,\text{and}\phantom{\rule{0.2em}{0ex}}c,\hfill \\ \begin{assortment}{cccccc}& & \text{if}\hfill & \hfill a& <\hfill & b\hfill \\ & & \text{so}\hfill & \hfill a-c& <\hfill & b-c.\hfill \\ \\ & & \text{if}\hfill & \hfill a& >\hfill & b\hfill \\ & & \text{then}\hfill & \hfill a-c& >\hfill & b-c.\hfill \end{array}\hfill & & & \begin{array}{cccccc}& & \text{if}\hfill & \hfill a& <\hfill & b\hfill \\ & & \text{then}\hfill & \hfill a+c& <\hfill & b+c.\hfill \\ \\ & & \text{if}\hfill & \hfill a& >\hfill & b\hfill \\ & & \text{then}\hfill & \hfill a+c& >\hfill & b+c.\hfill \end{array}\hfill \end{array}$

We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality $x+5>9$ , the steps would look like this:

	$x+5>9$
Decrease v from both sides to isolate $x$ .	$x+5-5>9-5$
Simplify.	$x>4$

Whatsoever number greater than 4 is a solution to this inequality.

Solve the inequality $n-\frac{1}{2}\le \frac{5}{8}$ , graph the solution on the number line, and write the solution in interval notation.

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$p-\frac{3}{4}\ge \frac{1}{6}$

This figure shows the inequality p is greater than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at p equals 11/12, and a dark line extends to the right from 11/12. Below the number line is the solution written in interval notation: bracket, 11/12 comma infinity, parenthesis.

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$r-\frac{1}{3}\le \frac{7}{12}$

This figure shows the inequality r is less than or equal to 11/12. Below this inequality is the inequality graphed on a number line ranging from 0 to 4, with tick marks at each integer. There is a bracket at r equals 11/12, and a dark line extends to the left from 11/12. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/12, bracket.

Solve Inequalities using the Division and Multiplication Backdrop of Inequality

The Division and Multiplication Backdrop of Equality land that if two quantities are equal, when we divide or multiply both quantities past the same amount, the results volition also be equal (provided we don't divide by 0).

Properties of Equality

$\begin{array}{cccc}\mathbf{\text{Division Property of Equality}}\hfill & & & \mathbf{\text{Multiplication Property of Equality}}\hfill \\ \text{For any numbers}\phantom{\rule{0.2em}{0ex}}a,b,c,\text{and}\phantom{\rule{0.2em}{0ex}}c\ne 0,\hfill & & & \text{For any real numbers}\phantom{\rule{0.2em}{0ex}}a,b,c,\hfill \\ \phantom{\rule{1em}{0ex}}\begin{array}{cccc}\text{if}\hfill & a\hfill & =\hfill & b,\hfill \\ \text{then}\hfill & \frac{a}{c}\hfill & =\hfill & \frac{b}{c}.\hfill \end{array}\hfill & & & \phantom{\rule{1em}{0ex}}\begin{array}{cccc}\text{if}\hfill & a\hfill & =\hfill & b,\hfill \\ \text{then}\hfill & ac\hfill & =\hfill & bc.\hfill \end{array}\hfill \end{array}$

Are there similar properties for inequalities? What happens to an inequality when nosotros divide or multiply both sides by a abiding?

Consider some numerical examples.


Divide both sides by five.		Multiply both sides by five.
Simplify.
Fill in the inequality signs.

$\mathbf{\text{The inequality signs stayed the same.}}$

Does the inequality stay the same when we divide or multiply by a negative number?


Divide both sides by −v.		Multiply both sides by −v.
Simplify.
Fill up in the inequality signs.

$\mathbf{\text{The inequality signs reversed their direction.}}$

When we divide or multiply an inequality by a positive number, the inequality sign stays the aforementioned. When we split or multiply an inequality past a negative number, the inequality sign reverses.

Here are the Sectionalisation and Multiplication Backdrop of Inequality for easy reference.

Partition and Multiplication Properties of Inequality

$\begin{array}{}\\ \\ \\ \text{For any real numbers}\phantom{\rule{0.2em}{0ex}}a,b,c\hfill \\ \\ \\ \\ \brainstorm{array}{c}\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a<b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c>0,\text{and then}\phantom{\rule{1em}{0ex}}\frac{a}{c}<\frac{b}{c}\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}air conditioning<bc.\hfill \\ \phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a>b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c>0,\text{then}\phantom{\rule{1em}{0ex}}\frac{a}{c}>\frac{b}{c}\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}ac>bc.\hfill \\ \phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a<b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c<0,\text{then}\phantom{\rule{1em}{0ex}}\frac{a}{c}>\frac{b}{c}\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}ac>bc.\hfill \\ \phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{0.2em}{0ex}}a>b\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}c<0,\text{then}\phantom{\dominion{1em}{0ex}}\frac{a}{c}<\frac{b}{c}\phantom{\rule{0.3em}{0ex}}\text{and}\phantom{\dominion{0.2em}{0ex}}ac<bc.\hfill \end{array}\hfill \end{array}$

When we divide or multiply an inequality past a:

positive number, the inequality stays the same.
negative number, the inequality reverses.

Solve the inequality $7y<\text{}\text{}42$ , graph the solution on the number line, and write the solution in interval note.

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$\left(8,\infty \right)$

$c>8$

This figure is a number line ranging from 6 to 10 with tick marks for each integer. The inequality c is greater than 8 is graphed on the number line, with an open parenthesis at c equals 8, and a dark line extending to the right of the parenthesis.

Solve the inequality, graph the solution on the number line, and write the solution in interval annotation.

$12d\le \text{}60$

$\left(-\infty ,5\right]$

This figure is a number line ranging from 3 to 7 with tick marks for each integer. The inequality d is less than or equal to 5 is graphed on the number line, with an open bracket at d equals 5, and a dark line extending to the left of the bracket. The inequality is also written in interval notation as parenthesis, negative infinity comma 5, bracket.

Solve the inequality $-10a\ge 50$ , graph the solution on the number line, and write the solution in interval note.

Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$-8q<32$

$q>-4$

This figure is a number line ranging from negative 6 to negative 3 with tick marks for each integer. The inequality q is greater than negative 4 is graphed on the number line, with an open parenthesis at q equals negative 4, and a dark line extending to the right of the parenthesis. The inequality is also written in interval notation as parenthesis, negative 4 comma infinity, parenthesis.

Solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$-7r\le \text{}-70$

This figure is a number line ranging from 9 to 13 with tick marks for each integer. The inequality r is greater than or equal to 10 is graphed on the number line, with an open bracket at r equals 10, and a dark line extending to the right of the bracket. The inequality is also written in interval notation as bracket, 10 comma infinity, parenthesis.

Solving Inequalities

Sometimes when solving an inequality, the variable ends up on the right. Nosotros can rewrite the inequality in reverse to get the variable to the left.

$\begin{array}{ccc}& & x>a\phantom{\dominion{0.2em}{0ex}}\text{has the aforementioned meaning equally}\phantom{\dominion{0.2em}{0ex}}a<x\hfill \end{array}$

Think nearly it equally "If Xavier is taller than Alex, then Alex is shorter than Xavier."

Solve the inequality $-20<\frac{4}{5}u$ , graph the solution on the number line, and write the solution in interval notation.

Solve the inequality, graph the solution on the number line, and write the solution in interval note.

$24\le \frac{3}{8}m$

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$-24<\frac{4}{3}n$

Solve the inequality $\frac{t}{-2}\ge 8$ , graph the solution on the number line, and write the solution in interval notation.

Solve the inequality, graph the solution on the number line, and write the solution in interval notation.

$\frac{k}{-12}\le 15$

Solve the inequality, graph the solution on the number line, and write the solution in interval annotation.

$\frac{u}{-4}\ge -16$

Solve Inequalities That Require Simplification

Most inequalities will take more i step to solve. We follow the same steps we used in the general strategy for solving linear equations, only be sure to pay close attention during multiplication or partition.

Solve the inequality $4m\le 9m+17$ , graph the solution on the number line, and write the solution in interval note.

Solve the inequality $3q\text{\hspace{0.17em}}\ge \text{\hspace{0.17em}}7q\text{\hspace{0.17em}}-\text{\hspace{0.17em}}23$ , graph the solution on the number line, and write the solution in interval notation.

Solve the inequality , graph the solution on the number line, and write the solution in interval notation.

Solve the inequality $8p+3\left(p-12\right)>7p-28$ , graph the solution on the number line, and write the solution in interval note.

Solve the inequality $9y+2\left(y+6\right)>5y-24$ , graph the solution on the number line, and write the solution in interval annotation.

Solve the inequality $6u+8\left(u-1\right)>10u+32$ , graph the solution on the number line, and write the solution in interval notation.

Merely similar some equations are identities and some are contradictions, inequalities may be identities or contradictions, as well. We recognize these forms when we are left with but constants as we solve the inequality. If the result is a true argument, nosotros have an identity. If the result is a false argument, nosotros take a contradiction.

Solve the inequality $8x-2\left(5-x\correct)<4\left(x+9\right)+6x$ , graph the solution on the number line, and write the solution in interval notation.

Solve the inequality $4b-3\left(3-b\right)>5\left(b-6\right)+2b$ , graph the solution on the number line, and write the solution in interval notation.

This figure shows an inequality that is an identity. Below this inequality is a number line ranging from negative 2 to 2 with tick marks for each integer. The identity is graphed on the number line, with a dark line extending in both directions. The inequality is also written in interval notation as parenthesis, negative infinity comma infinity, parenthesis.

Solve the inequality $9h-7\left(2-h\right)<8\left(h+11\right)+8h$ , graph the solution on the number line, and write the solution in interval note.

Solve the inequality $\frac{1}{3}a-\frac{1}{8}a>\frac{5}{24}a\text{}+\frac{3}{4}$ , graph the solution on the number line, and write the solution in interval notation.

Solve the inequality $\frac{1}{4}x-\frac{1}{12}x>\frac{1}{6}x+\frac{7}{8}$ , graph the solution on the number line, and write the solution in interval notation.

This figure shows an inequality that is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. No inequality is graphed on the number line. Below the number line is the statement:

Solve the inequality $\frac{2}{5}z-\frac{1}{3}z<\frac{1}{15}z\text{}-\frac{3}{5}$ , graph the solution on the number line, and write the solution in interval annotation.

Interpret to an Inequality and Solve

To interpret English sentences into inequalities, we need to recognize the phrases that signal the inequality. Some words are easy, like 'more than than' and 'less than'. But others are non equally obvious.

Think about the phrase 'at least' – what does it mean to be 'at least 21 years old'? It means 21 or more. The phrase 'at least' is the same as 'greater than or equal to'.

(Effigy) shows some common phrases that indicate inequalities.

$>$	$\ge$	$<$	$\le$
is greater than	is greater than or equal to	is less than	is less than or equal to
is more than	is at to the lowest degree	is smaller than	is at most
is larger than	is no less than	has fewer than	is no more than
exceeds	is the minimum	is lower than	is the maximum

Translate and solve. Then write the solution in interval annotation and graph on the number line.

Twelve times c is no more than than 96.

Translate and solve. So write the solution in interval notation and graph on the number line.

Twenty times y is at most 100

Translate and solve. Then write the solution in interval notation and graph on the number line.

Nine times z is no less than 135

Translate and solve. And then write the solution in interval notation and graph on the number line.

Thirty less than 10 is at to the lowest degree 45.

Translate and solve. And then write the solution in interval notation and graph on the number line.

19 less than p is no less than 47

Translate and solve. Then write the solution in interval notation and graph on the number line.

Four more a is at most 15.

Fundamental Concepts

Subtraction Property of Inequality
For any numbers a, b, and c,
if $a<b$ then and
if $a>b$ then $a-c>b-c.$
Improver Property of Inequality
For any numbers a, b, and c,
if then $a+c<b+c$ and
if $a>b$ then $a+c>b+c.$
Division and Multiplication Properties of Inequalit y
For any numbers a, b, and c,
if $a<b$ and $c>0$ , then $\frac{a}{c}<\frac{b}{c}$ and $ac>bc$ .
if $a>b$ and $c>0$ , then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$ .
if and , and then $\frac{a}{c}>\frac{b}{c}$ and $ac>bc$ .
if $a>b$ and $c<0$ , then $\frac{a}{c}<\frac{b}{c}$ and $air conditioning<bc$ .
When we divide or multiply an inequality by a:
- positive number, the inequality stays the same.
- negative number, the inequality reverses.

Section Exercises

Practice Makes Perfect

Graph Inequalities on the Number Line

In the post-obit exercises, graph each inequality on the number line.

In the following exercises, graph each inequality on the number line and write in interval note.

Solve Inequalities using the Subtraction and Addition Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$n-11<33$

$m-45\le 62$

$u+25>21$

$v+12>3$

$a+\frac{3}{4}\ge \frac{7}{10}$

$b+\frac{7}{8}\ge \frac{1}{6}$

$f-\frac{13}{20}<-\frac{5}{12}$

$g-\frac{eleven}{12}<-\frac{5}{18}$

Solve Inequalities using the Segmentation and Multiplication Properties of Inequality

In the post-obit exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$8x>72$

$7r\le 56$

$9s\ge 81$

$-5u\ge 65$

$-8v\le 96$

$-9c<126$

$-7d>105$

$20>\frac{2}{5}h$

$forty<\frac{5}{8}k$

$\frac{7}{6}j\ge 42$

$\frac{9}{4}g\le 36$

$\frac{a}{-3}\le 9$

$\frac{b}{-10}\ge 30$

$-25<\frac{p}{-5}$

$-18>\frac{q}{-6}$

$9t\ge -27$

$\frac{2}{3}y>-36$

$\frac{3}{5}x\le -45$

Solve Inequalities That Require Simplification

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$4v\ge 9v-40$

$5u\le 8u-21$

$9p>14p-18$

$12x+3\left(x+7\right)>10x-24$

$9y+5\left(y+3\correct)<4y-35$

$6h-4\left(h-1\right)\le 7h-11$

$4k-\left(k-2\right)\ge 7k-26$

$8m-2\left(14-m\right)\ge \text{}7\left(m-4\right)+3m$

$6n-12\left(3-n\right)\le 9\left(n-4\right)+9n$

$\frac{iii}{4}b-\frac{1}{three}b<\frac{5}{12}b-\frac{1}{2}$

$9u+5\left(2u-5\right)\ge 12\left(u-1\right)+7u$

At the top of this figure is the result of the inequality: the inequality is a contradiction. Below this is a number line ranging from negative 2 to 2 with tick marks for each integer. Because this is a contradiction, no inequality is graphed on the number line. Below the number line is the statement:

$\frac{2}{3}g-\frac{1}{2}\left(g-14\right)\le \frac{1}{6}\left(g+42\right)$

$\frac{5}{6}a-\frac{1}{4}a>\frac{7}{12}a+\frac{2}{3}$

$\frac{4}{5}h-\frac{2}{3}\left(h-9\right)\ge \frac{1}{15}\left(2h+90\right)$

$12v+3\left(4v-1\right)\le 19\left(v-2\right)+5v$

Mixed practice

In the post-obit exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$15k\le -40$

$35k\ge -77$

$23p-2\left(6-5p\right)>3\left(11p-4\right)$

$18q-4\left(10-3q\right)<5\left(6q-8\right)$

$-\frac{9}{4}x\ge -\frac{5}{12}$

$-\frac{21}{8}y\le -\frac{15}{28}$

$c+34<-99$

$d+29>-61$

$\frac{m}{18}\ge -4$

$\frac{n}{13}\le -6$

Translate to an Inequality and Solve

In the following exercises, interpret and solve .Then write the solution in interval notation and graph on the number line.

Fourteen times d is greater than 56.

Ninety times c is less than 450.

Eight times z is smaller than $-40$ .

Ten times y is at almost $-110$ .

Three more than h is no less than 25.

Six more thousand exceeds 25.

Ten less than w is at least 39.

Twelve less than x is no less than 21.

Negative v times r is no more than 95.

Negative two times s is lower than 56.

Xix less than b is at nigh $-22$ .

15 less than a is at least $-7$ .

Everyday Math

Prophylactic A child'due south summit, h, must be at least 57 inches for the child to safely ride in the front seat of a car. Write this every bit an inequality.

Fighter pilots The maximum pinnacle, h, of a fighter airplane pilot is 77 inches. Write this as an inequality.

$h\le 77$

Elevators The total weight, west, of an elevator's passengers tin exist no more than than 1,200 pounds. Write this as an inequality.

Shopping The number of items, n, a shopper can have in the express check-out lane is at near 8. Write this as an inequality.

$n\le 8$

Writing Exercises

Give an example from your life using the phrase 'at to the lowest degree'.

Give an example from your life using the phrase 'at nearly'.

Answers will vary.

Explain why it is necessary to opposite the inequality when solving $-5x>10$ .

Explain why it is necessary to reverse the inequality when solving $\frac{n}{-3}<12$ .

Answers volition vary.

Cocky Check

ⓐ Afterward completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right:

ⓑ What does this checklist tell you lot about your mastery of this department? What steps will you lot have to improve?

Chapter 2 Review Exercises

Solve Equations using the Subtraction and Addition Backdrop of Equality

Verify a Solution of an Equation

In the post-obit exercises, determine whether each number is a solution to the equation.

$10x-1=5x;x=\frac{1}{5}$

$w+2=\frac{5}{8};w=\frac{3}{8}$

$-12n+5=8n;n=-\frac{5}{4}$

$6a-3=-7a,a=\frac{3}{13}$

aye

Solve Equations using the Subtraction and Improver Backdrop of Equality

In the post-obit exercises, solve each equation using the Subtraction Property of Equality.

$x+7=19$

$y+2=-6$

$y=-8$

$a+\frac{1}{3}=\frac{5}{3}$

$n+3.6=5.1$

$n=1.5$

In the following exercises, solve each equation using the Addition Property of Equality.

$u-7=10$

$x-9=-4$

$x=5$

$c-\frac{3}{11}=\frac{9}{11}$

$p-4.8=14$

$p=18.8$

In the post-obit exercises, solve each equation.

$n-12=32$

$y+16=-9$

$y=-25$

$f+\frac{2}{3}=4$

$d-3.9=8.2$

$d=12.1$

Solve Equations That Require Simplification

In the following exercises, solve each equation.

$y+8-15=-3$

$7x+10-6x+3=5$

$x=-8$

$6\left(n-1\right)-5n=-14$

$8\left(3p+5\right)-23\left(p-1\right)=35$

$p=-28$

Translate to an Equation and Solve

In the following exercises, translate each English sentence into an algebraic equation and so solve it.

The sum of $-6$ and $m$ is 25.

Four less than $n$ is 13.

$n-4=13;n=17$

Translate and Solve Applications

In the post-obit exercises, translate into an algebraic equation and solve.

Rochelle'southward daughter is 11 years old. Her son is three years younger. How quondam is her son?

Tan weighs 146 pounds. Minh weighs 15 pounds more than Tan. How much does Minh counterbalance?

161 pounds

Peter paid ?9.75 to become to the movies, which was ?46.25 less than he paid to go to a concert. How much did he pay for the concert?

Elissa earned ?152.84 this calendar week, which was ?21.65 more than she earned last week. How much did she earn last week?

?131.19

Solve Equations using the Division and Multiplication Properties of Equality

Solve Equations Using the Division and Multiplication Properties of Equality

In the following exercises, solve each equation using the sectionalization and multiplication properties of equality and cheque the solution.

$8x=72$

$13a=-65$

$a=-5$

$0.25p=5.25$

$\text{−}y=4$

$y=-4$

$\frac{n}{6}=18$

$\frac{y}{-10}=30$

$y=-300$

$36=\frac{3}{4}x$

$\frac{5}{8}u=\frac{15}{16}$

$u=\frac{3}{2}$

$-18m=-72$

$\frac{c}{9}=36$

$c=324$

$0.45x=6.75$

$\frac{11}{12}=\frac{2}{3}y$

$y=\frac{11}{8}$

Solve Equations That Require Simplification

In the following exercises, solve each equation requiring simplification.

$5r-3r+9r=35-2$

$24x+8x-11x=-7-14$

$x=-1$

$\frac{11}{12}n-\frac{5}{6}n=9-5$

$-9\left(d-2\right)-15=-24$

$d=3$

Translate to an Equation and Solve

In the post-obit exercises, interpret to an equation and and then solve.

143 is the product of $-11$ and y.

The caliber of b and and nine is $-27$ .

$\frac{b}{9}=-27;b=-243$

The sum of q and one-fourth is one.

The difference of due south and ane-twelfth is one 4th.

$s-\frac{1}{12}=\frac{1}{4};s=\frac{1}{3}$

Translate and Solve Applications

In the following exercises, interpret into an equation and solve.

Ray paid ?21 for 12 tickets at the county fair. What was the toll of each ticket?

Janet gets paid ?24 per hour. She heard that this is $\frac{3}{4}$ of what Adam is paid. How much is Adam paid per hour?

?32

Solve Equations with Variables and Constants on Both Sides

Solve an Equation with Constants on Both Sides

In the following exercises, solve the following equations with constants on both sides.

$8p+7=47$

$10w-5=65$

$w=7$

$3x+19=-47$

$32=-4-9n$

$n=-4$

Solve an Equation with Variables on Both Sides

In the following exercises, solve the following equations with variables on both sides.

$7y=6y-13$

$5a+21=2a$

$a=-7$

$k=-6k-35$

$4x-\frac{3}{8}=3x$

$x=\frac{3}{8}$

Solve an Equation with Variables and Constants on Both Sides

In the post-obit exercises, solve the following equations with variables and constants on both sides.

$12x-9=3x+45$

$5n-20=-7n-80$

$n=-5$

$4u+16=-19-u$

$\frac{5}{8}c-4=\frac{3}{8}c+4$

$c=32$

Use a Full general Strategy for Solving Linear Equations

Solve Equations Using the General Strategy for Solving Linear Equations

In the following exercises, solve each linear equation.

$6\left(x+6\right)=24$

$9\left(2p-5\right)=72$

$p=\frac{13}{2}$

$\text{−}\left(s+4\right)=18$

$8+3\left(n-9\right)=17$

$n=12$

$23-3\left(y-7\right)=8$

$\frac{1}{3}\left(6m+21\right)=m-7$

$m=-14$

$4\left(3.5y+0.25\right)=365$

$0.25\left(q-8\right)=0.1\left(q+7\right)$

$q=18$

$8\left(r-2\right)=6\left(r+10\right)$

$x=-1$

$k=\frac{3}{4}$

Allocate Equations

In the following exercises, allocate each equation as a conditional equation, an identity, or a contradiction so state the solution.

contradiction; no solution

$-8\left(7m+4\right)=-6\left(8m+9\right)$

identity; all real numbers

Solve Equations with Fractions and Decimals

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

$\frac{2}{5}n-\frac{1}{10}=\frac{7}{10}$

$\frac{1}{3}x+\frac{1}{5}x=8$

$x=15$

$\frac{3}{4}a-\frac{1}{3}=\frac{1}{2}a-\frac{5}{6}$

$\frac{1}{2}\left(k-3\right)=\frac{1}{3}\left(k+16\right)$

$k=41$

$\frac{3x-2}{5}=\frac{3x+4}{8}$

$\frac{5y-1}{3}+4=\frac{-8y+4}{6}$

$y=-1$

Solve Equations with Decimal Coefficients

In the post-obit exercises, solve each equation with decimal coefficients.

$0.8x-0.3=0.7x+0.2$

$0.36u+2.55=0.41u+6.8$

$u=-85$

$0.6p-1.9=0.78p+1.7$

$d=-20$

Solve a Formula for a Specific Variable

Utilize the Altitude, Rate, and Fourth dimension Formula

In the following exercises, solve.

Natalie drove for $7\frac{1}{2}$ hours at 60 miles per hour. How much distance did she travel?

Mallory is taking the bus from St. Louis to Chicago. The altitude is 300 miles and the motorbus travels at a steady rate of 60 miles per hour. How long will the bus ride be?

5 hours

Aaron'due south friend drove him from Buffalo to Cleveland. The altitude is 187 miles and the trip took two.75 hours. How fast was Aaron's friend driving?

Link rode his bike at a steady charge per unit of 15 miles per hour for $2\frac{1}{2}$ hours. How much distance did he travel?

37.5 miles

Solve a Formula for a Specific Variable

In the following exercises, solve.

Solve the formula $4x+3y=6$ for y
ⓐ when $x=-2$
ⓑ in general

ⓐ $y=\frac{14}{3}$ ⓑ $y=\frac{6-4x}{3}$

Solve $180=a+b+c$ for $c$ .

Solve Linear Inequalities

Graph Inequalities on the Number Line

In the post-obit exercises, graph each inequality on the number line.

In the following exercises, graph each inequality on the number line and write in interval notation.

Solve Inequalities using the Subtraction and Addition Properties of Inequality

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$n-12\le 23$

$m+14\le 56$

$a+\frac{2}{3}\ge \frac{7}{12}$

$b-\frac{7}{8}\ge -\frac{1}{2}$

Solve Inequalities using the Sectionalisation and Multiplication Properties of Inequality

In the post-obit exercises, solve each inequality, graph the solution on the number line, and write the solution in interval annotation.

$9x>54$

$-12d\le 108$

$\frac{5}{2}j<-60$

$\frac{q}{-2}\ge -24$

Solve Inequalities That Require Simplification

In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.

$6p>15p-30$

$9h-7\left(h-1\right)\le 4h-23$

$5n-15\left(4-north\right)<10\left(n-6\right)+10n$

$\frac{3}{8}a-\frac{1}{12}a>\frac{5}{12}a+\frac{3}{4}$

Translate to an Inequality and Solve

In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.

5 more than z is at most 19.

Three less than c is at least 360.

Negative 2 times a is no more 8.

Everyday Math

Describe how you have used two topics from this chapter in your life outside of your math class during the past calendar month.

Chapter ii Practice Examination

Make up one's mind whether each number is a solution to the equation $6x-3=x+20$ .

ⓐ 5
ⓑ $\frac{23}{5}$

ⓐ no ⓑ aye

In the following exercises, solve each equation.

$n-\frac{2}{3}=\frac{1}{4}$

$\frac{9}{2}c=144$

$c=32$

$4y-8=16$

$-8x-15+9x-1=-21$

$x=-5$

$-15a=120$

$\frac{2}{3}x=6$

$x=9$

$x-3.8=8.2$

$10y=-5y-60$

$y=-4$

$8n-2=6n-12$

$9m-2-4m-m=42-8$

$m=9$

$-5\left(2x-1\right)=45$

$\text{−}\left(d-9\right)=23$

$d=-14$

$\frac{1}{4}\left(12m-28\right)=6-2\left(3m-1\right)$

$2\left(6x-5\right)-8=-22$

$x=-\frac{1}{3}$

$8\left(3a-5\right)-7\left(4a-3\right)=20-3a$

$\frac{1}{4}p-\frac{1}{3}=\frac{1}{2}$

$p=\frac{10}{3}$

$0.1d+0.25\left(d+8\right)=4.1$

$14n-3\left(4n+5\right)=-9+2\left(n-8\right)$

contradiction; no solution

$9\left(3u-2\right)-4\left[6-8\left(u-1\right)\right]$ $=3\left(u-2\right)$

Solve the formula $x-2y=5$ for y
ⓐ when $x=-3$
ⓑ in full general

ⓐ $y=4$ ⓑ $y=\frac{5-x}{2}$

In the post-obit exercises, graph on the number line and write in interval notation.

$x\ge -3.5$

$x<\frac{11}{4}$

This figure is a number line ranging from 1 to 5 with tick marks for each integer. The inequality x is less than 11/4 is graphed on the number line, with an open parenthesis at x equals 11/4, and a dark line extending to the left of the parenthesis. Below the number line is the solution written in interval notation: parenthesis, negative infinity comma 11/4, parenthesis.

In the post-obit exercises,, solve each inequality, graph the solution on the number line, and write the solution in interval note.

$8k\ge 5k-120$

$3c-10\left(c-2\correct)<5c+16$

In the following exercises, interpret to an equation or inequality and solve.

4 less than twice ten is sixteen.

Fifteen more than than n is at least 48.

$n+15\ge 48;n\ge 33$

Samuel paid ?25.82 for gas this week, which was ?3.47 less than he paid terminal calendar week. How much had he paid final week?

Jenna bought a coat on sale for ?120, which was $\frac{2}{3}$ of the original price. What was the original price of the coat?

$120=\frac{2}{3}p$ ; The original price was ?180.

Sean took the bus from Seattle to Boise, a distance of 506 miles. If the trip took $7\frac{2}{3}$ hours, what was the speed of the bus?

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Source: https://opentextbc.ca/elementaryalgebraopenstax/chapter/solve-linear-inequalities/

what is the solution to the inequality? 6xÃ¢Ë†â€™5>Ã¢Ë†â€™29

23 Solve Linear Inequalities

Learning Objectives

Graph Inequalities on the Number Line

Solve Inequalities using the Subtraction and Add-on Properties of Inequality

Solve Inequalities using the Division and Multiplication Backdrop of Inequality

Solve Inequalities That Require Simplification

Interpret to an Inequality and Solve

Fundamental Concepts

Section Exercises

Practice Makes Perfect

Everyday Math

Writing Exercises

Cocky Check

Chapter 2 Review Exercises

Solve Equations using the Subtraction and Addition Backdrop of Equality

Solve Equations using the Division and Multiplication Properties of Equality

Solve Equations with Variables and Constants on Both Sides

Use a Full general Strategy for Solving Linear Equations

Solve Equations with Fractions and Decimals

Solve a Formula for a Specific Variable

Solve Linear Inequalities

Everyday Math

Chapter ii Practice Examination

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