Learning Objectives
By the end of this section, you will be able to:
 Graph inequalities on the number line
 Solve inequalities using the Subtraction and Addition Properties of inequality
 Solve inequalities using the Division and Multiplication Properties of inequality
 Solve inequalities that require simplification
 Translate to an inequality and solve
Be Prepared 2.18
Before you get started, take this readiness quiz.
Translate from algebra to English: $15>x$.
If you missed this problem, review Example 1.12.
Be Prepared 2.19
Solve: $n9=\mathrm{42}.$
If you missed this problem, review Example 2.3.
Be Prepared 2.20
Solve: $\mathrm{5}p=\mathrm{23}.$
If you missed this problem, review Example 2.13.
Be Prepared 2.21
Solve: $3a12=7a20.$
If you missed this problem, review Example 2.34.
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 statement when substituted into the equation.
What about the solution of an inequality? What number would make the inequality $x>3$ true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 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 3, to show that all numbers greater than 3 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 2.7. Please note that the following convention is used: thick arrows point in the positive direction and thin arrows point in the negative direction.
The graph of the inequality $x\ge 3$ is very much like the graph of $x>3$, but now we need to show that 3 is a solution, too. We do that by putting a bracket at $x=3$, as shown in Figure 2.8.
Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.
Example 2.66
Graph on the number line:
ⓐ $x\le 1$ ⓑ $x<5$ ⓒ $x>1$
Solution
 ⓐ $x\le 1$
This means all numbers less than or equal to 1. We shade in all the numbers on the number line to the left of 1 and put a bracket at $x=1$ to show that it is included.
 ⓑ $x<5$
This means all numbers less than 5, but not including 5. We shade in all the numbers on the number line to the left of 5 and put a parenthesis at $x=5$ to show it is not included.
 ⓒ $x>1$
This means all numbers greater than $\mathrm{1}$, but not including $\mathrm{1}$. We shade in all the numbers on the number line to the right of $\mathrm{1}$, then put a parenthesis at $x=\mathrm{1}$ to show it is not included.
Try It 2.131
Graph on the number line: ⓐ $x\le 1$ ⓑ $x>2$ ⓒ $x<3$
Try It 2.132
Graph on the number line: ⓐ $x>2$ ⓑ $x<3$ ⓒ $x\ge \mathrm{1}$
We can also represent inequalities using interval notation. As we saw above, the inequality $x>3$ means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express $x>3$ as $\left(3,\infty \right).$ The symbol $\infty $ is read as ‘infinity’. It is not an actual number. Figure 2.9 shows both the number line and the interval notation.
The inequality $x\le 1$ means all numbers less than or equal to 1. There is no lower end to those numbers. We write $x\le 1$ in interval notation as $(\text{\u2212}\infty ,1]$. The symbol $\text{\u2212}\infty $ is read as ‘negative infinity’. Figure 2.10 shows both the number line and interval notation.
Inequalities, Number Lines, and Interval Notation
Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in Figure 2.11.
Example 2.67
Graph on the number line and write in interval notation.
ⓐ $x\ge \mathrm{3}$ ⓑ $x<2.5$ ⓒ $x\le \frac{3}{5}$
Solution
 ⓐ
Shade to the right of $\mathrm{3}$, and put a bracket at $\mathrm{3}$. Write in interval notation.
ⓑShade to the left of $2.5$, and put a parenthesis at $2.5$. Write in interval notation.
ⓒShade to the left of $\frac{3}{5}$, and put a bracket at $\frac{3}{5}$. Write in interval notation.
Try It 2.133
Graph on the number line and write in interval notation:
ⓐ $x>2$ ⓑ $x\le 1.5$ ⓒ $x\ge \frac{3}{4}$
Try It 2.134
Graph on the number line and write in interval notation:
ⓐ $x\le 4$ ⓑ $x\ge 0.5$ ⓒ $x<\frac{2}{3}$
Solve Inequalities using the Subtraction and Addition 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 will be equal.
Properties of Equality
Similar properties hold true for inequalities.
For example, we know that −4 is less than 2.  
If we subtract 5 from both quantities, is the left side still less than the right side? 

We get −9 on the left and −3 on the right.  
And we know −9 is less than −3.  
The inequality sign stayed the same. 
Similarly we could show that the inequality also stays the same for addition.
This leads us to the Subtraction and Addition Properties of Inequality.
Properties of Inequality
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:
$x+5>9$  
Subtract 5 from both sides to isolate $x$.  $x+55>95$ 
Simplify.  $x>4$ 
Any number greater than 4 is a solution to this inequality.
Example 2.68
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.
Solution
Add $\frac{1}{2}$ to both sides of the inequality.  
Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation.  $(\infty ,\frac{9}{8}]$ 
Try It 2.135
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
$p\frac{3}{4}\ge \frac{1}{6}$
Try It 2.136
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
$r\frac{1}{3}\le \frac{7}{12}$
Solve Inequalities using the Division and Multiplication Properties of Inequality
The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).
Properties of Equality
Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?
Consider some numerical examples.
Divide both sides by 5.  Multiply both sides by 5.  
Simplify.  
Fill in the inequality signs. 
Does the inequality stay the same when we divide or multiply by a negative number?
Divide both sides by −5.  Multiply both sides by −5.  
Simplify.  
Fill in the inequality signs. 
When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.
Here are the Division and Multiplication Properties of Inequality for easy reference.
Division and Multiplication Properties of Inequality
When we divide or multiply an inequality by a:
 positive number, the inequality stays the same.
 negative number, the inequality reverses.
Example 2.69
Solve the inequality $7y<\text{}\text{}42$, graph the solution on the number line, and write the solution in interval notation.
Solution
Divide both sides of the inequality by 7. Since $7>0$, the inequality stays the same. 

Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Try It 2.137
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
$c8>0$
Try It 2.138
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
$12d\le \text{}60$
Example 2.70
Solve the inequality $\mathrm{10}a\ge 50$, graph the solution on the number line, and write the solution in interval notation.
Solution
Divide both sides of the inequality by −10. Since $\mathrm{10}<0$, the inequality reverses. 

Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Try It 2.139
Solve each inequality, graph the solution on the number line, and write the solution in interval notation.
$\mathrm{8}q<32$
Try It 2.140
Solve each inequality, graph the solution on the number line, and write the solution in interval notation.
$\mathrm{7}r\le \text{}70$
Solving Inequalities
Sometimes when solving an inequality, the variable ends up on the right. We can rewrite the inequality in reverse to get the variable to the left.
Think about it as “If Xavier is taller than Alex, then Alex is shorter than Xavier.”
Example 2.71
Solve the inequality $\mathrm{20}<\frac{4}{5}u$, graph the solution on the number line, and write the solution in interval notation.
Solution
Multiply both sides of the inequality by $\frac{5}{4}$. Since $\frac{5}{4}>0$, the inequality stays the same. 

Simplify.  
Rewrite the variable on the left.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Try It 2.141
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
$24\le \frac{3}{8}m$
Try It 2.142
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
$\mathrm{24}<\frac{4}{3}n$
Example 2.72
Solve the inequality $\frac{t}{\mathrm{2}}\ge 8$, graph the solution on the number line, and write the solution in interval notation.
Solution
Multiply both sides of the inequality by $\mathrm{2}$. Since $\mathrm{2}<0$, the inequality reverses. 

Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Try It 2.143
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
$\frac{k}{\mathrm{12}}\le 15$
Try It 2.144
Solve the inequality, graph the solution on the number line, and write the solution in interval notation.
$\frac{u}{\mathrm{4}}\ge \mathrm{16}$
Solve Inequalities That Require Simplification
Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.
Example 2.73
Solve the inequality $4m\le 9m+17$, graph the solution on the number line, and write the solution in interval notation.
Solution
Subtract $9m$ from both sides to collect the variables on the left.  
Simplify.  
Divide both sides of the inequality by −5, and reverse the inequality.  
Simplify.  
Graph the solution on the number line.  
Write the solution in interval notation. 
Try It 2.145
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.
Try It 2.146
Solve the inequality $6x<10x+19$, graph the solution on the number line, and write the solution in interval notation.
Example 2.74
Solve the inequality $8p+3(p12)>7p28$, graph the solution on the number line, and write the solution in interval notation.
Solution
Simplify each side as much as possible.  $8p+3(p12)>7p28$ 
Distribute.  $\phantom{\rule{0.6em}{0ex}}8p+3p36>7p28$ 
Combine like terms.  $\phantom{\rule{2.3em}{0ex}}11p36>7p28$ 
Subtract $7p$ from both sides to collect the variables on the left.  $\phantom{\rule{2.2em}{0ex}}11p367p>7p287p$ 
Simplify.  $\phantom{\rule{1.2em}{0ex}}4p36>\mathrm{28}$ 
Add 36 to both sides to collect the constants on the right.  $\phantom{\rule{1.3em}{0ex}}4p36+36>\mathrm{28}+36$ 
Simplify.  $\phantom{\rule{2em}{0ex}}4p>8$ 
Divide both sides of the inequality by 4; the inequality stays the same.  $\phantom{\rule{2em}{0ex}}\frac{4p}{4}>\frac{8}{4}$ 
Simplify.  $\phantom{\rule{2.5em}{0ex}}p>2$ 
Graph the solution on the number line.  
Write the solution in interal notation.  $(2,\mathrm{\infty})$ 
Try It 2.147
Solve the inequality $9y+2(y+6)>5y24$, graph the solution on the number line, and write the solution in interval notation.
Try It 2.148
Solve the inequality $6u+8(u1)>10u+32$, graph the solution on the number line, and write the solution in interval notation.
Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.
Example 2.75
Solve the inequality $8x2\left(5x\right)<4\left(x+9\right)+6x$, graph the solution on the number line, and write the solution in interval notation.
Solution
Simplify each side as much as possible.  $8x2(5x)<4(x+9)+6x$ 
Distribute.  $8x10+2x<4x+36+6x$ 
Combine like terms.  $10x10<10x+36$ 
Subtract $10x$ from both sides to collect the variables on the left.  $10x1010x<10x+3610x$ 
Simplify.  $\mathrm{10}<36\phantom{\rule{0.6em}{0ex}}$ 
The $x$’s are gone, and we have a true statement.  The inequality is an identity. The solution is all real numbers. 
Graph the solution on the number line.  
Write the solution in interval notation.  $(\mathrm{\infty},\mathrm{\infty})$ 
Try It 2.149
Solve the inequality $4b3(3b)>5(b6)+2b$, graph the solution on the number line, and write the solution in interval notation.
Try It 2.150
Solve the inequality $9h7\left(2h\right)<8\left(h+11\right)+8h$, graph the solution on the number line, and write the solution in interval notation.
Example 2.76
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.
Solution
Multiply both sides by the LCD, 24, to clear the fractions.  
Simplify.  
Combine like terms.  
Subtract $5a$ from both sides to collect the variables on the left.  
Simplify.  
The statement is false!  The inequality is a contradiction. 
There is no solution.  
Graph the solution on the number line.  
Write the solution in interval notation.  There is no solution. 
Try It 2.151
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.
Try It 2.152
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 notation.
Translate to an Inequality and Solve
To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as 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’.
Table 2.7 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 least  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 
Example 2.77
Translate and solve. Then write the solution in interval notation and graph on the number line.
Twelve times c is no more than 96.
Solution
Translate.  
Solve—divide both sides by 12.  
Simplify.  
Write in interval notation.  
Graph on the number line. 
Try It 2.153
Translate and solve. Then write the solution in interval notation and graph on the number line.
Twenty times y is at most 100
Try It 2.154
Translate and solve. Then write the solution in interval notation and graph on the number line.
Nine times z is no less than 135
Example 2.78
Translate and solve. Then write the solution in interval notation and graph on the number line.
Thirty less than x is at least 45.
Solution
Translate.  
Solve—add 30 to both sides.  
Simplify.  
Write in interval notation.  
Graph on the number line. 
Try It 2.155
Translate and solve. Then write the solution in interval notation and graph on the number line.
Nineteen less than p is no less than 47
Try It 2.156
Translate and solve. Then write the solution in interval notation and graph on the number line.
Four more than a is at most 15.
Section 2.7 Exercises
Practice Makes Perfect
Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line.
ⓐ $x\le 2$
ⓑ $x>1$ⓒ $x<0$
ⓐ $x\ge \mathrm{3}$ⓑ $x<4$ⓒ $x\le 2$
In the following exercises, graph each inequality on the number line and write in interval notation.
ⓐ $x<2$ⓑ $x\ge \mathrm{3.5}$ⓒ $x\le \frac{2}{3}$
ⓐ $x\ge \mathrm{4}$ⓑ $x<2.5$ⓒ $x>\frac{3}{2}$
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.
$n11<33$
$u+25>21$
$a+\frac{3}{4}\ge \frac{7}{10}$
$f\frac{13}{20}<\frac{5}{12}$
Solve Inequalities using the Division and Multiplication Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
$8x>72$
$7r\le 56$
$\mathrm{5}u\ge 65$
$\mathrm{9}c<126$
$20>\frac{2}{5}h$
$\frac{7}{6}j\ge 42$
$\frac{a}{\mathrm{3}}\le 9$
$\mathrm{25}<\frac{p}{\mathrm{5}}$
$9t\ge \mathrm{27}$
$\frac{2}{3}y>36$
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.
$4v\ge 9v40$
$13q<7q29$
$12x+3(x+7)>10x24$
$6h4(h1)\le 7h11$
$8m2\left(14m\right)\ge \text{}7\left(m4\right)+3m$
$\frac{3}{4}b\frac{1}{3}b<\frac{5}{12}b\frac{1}{2}$
$\frac{2}{3}g\frac{1}{2}(g14)\le \frac{1}{6}(g+42)$
$\frac{4}{5}h\frac{2}{3}\left(h9\right)\ge \frac{1}{15}\left(2h+90\right)$
Mixed practice
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
$15k\le 40$
$23p2(65p)>3(11p4)$
$\frac{9}{4}x\ge \frac{5}{12}$
$c+34<99$
$\frac{m}{18}\ge \mathrm{4}$
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.
Fourteen times d is greater than 56.
Eight times z is smaller than $\mathrm{40}$.
Three more than h is no less than 25.
Ten less than w is at least 39.
Negative five times r is no more than 95.
Nineteen less than b is at most $\mathrm{22}$.
Everyday Math
Safety A child’s height, h, must be at least 57 inches for the child to safely ride in the front seat of a car. Write this as an inequality.
Fighter pilots The maximum height, h, of a fighter pilot is 77 inches. Write this as an inequality.
Elevators The total weight, w, of an elevator’s passengers can be no more than 1,200 pounds. Write this as an inequality.
Shopping The number of items, n, a shopper can have in the express checkout lane is at most 8. Write this as an inequality.
Writing Exercises
Give an example from your life using the phrase ‘at least’.
Explain why it is necessary to reverse the inequality when solving $\mathrm{5}x>10$.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?