LINEAR INEQUALITIES
LESSON 2-A
A
mathematical sentence that contains >, <, ≥ or ≤ is an inequality. A solution to an inequality is any value
that makes the inequality true.
Jennifer has less than $75 in her bank account. Monte has at least 300 baseball cards in his collection. Each
of these statements can be written using an inequality. Inequalities are mathematical statements which use
>, <, ≥ or ≤ to show a relationship between quantities.
Jennifer has less than $75 in her bank account.
j
< $75
Monte has at least 300 baseball cards.
m ≥ 300
Inequalities have multiple answers that can make the statement true. In Jennifer’s example, she might have $2
or $74. All that is known for certain is that she has under $75 in her account. In Monte’s example, he might
have 300 cards (this comes from the "equal to" part of ≥) or 3,000,000 cards. There is an infinite number of
possibilities that would make this statement true.
EXAMPLE 1
Write an inequality for each statement.
a. Nanette’s height (h) is greater than 50 inches.
b. Pete has at least $6 in his pocket. Let p represent the amount of money.
c. Jeremy is less than 20 years old. Let j represent Jeremy’s age.
Solutions
a. The key words are “greater than”. Use the symbol >.
b. The key words are “at least”. This means the lowest
amount he could have is $6. Use the ≥ symbol.
h > 50
c. The key words are “less than”. Use the < symbol.
j < 20
2 Lesson 2-A ~ Linear Inequalities
p≥6
Solutions to an inequality can be graphed on a number line. When using the > or < inequality symbols, an
"open circle" is used on the number line because the solution does not include the given number. For example,
if x > 2, the solution cannot include 2 because 2 is not greater than 2. When using the ≥ or ≤ inequality
symbols, a "closed (or filled in) circle" is used because the solution contains the given number.
Determining which direction the arrow should point is based on the relationship
between the variable and the solution. When graphing an inequality on a number line,
the solution is shown by using an arrow to point towards the set of numbers that make
the statement true. Notice in the box below how the circle is either an open or closed
circle to designate the type of inequality that is being graphed.
Inequalities are solved using properties similar to those you used to solve equations. Use inverse operations to
isolate the variable so the solution can be graphed on a number line.
EXAMPLE 2
Solution
Solve the inequality and graph its solution on a number line.
3x − 1 < 14
Add 1 to both sides of the inequality.
3x − 1 < 14
+1 +1
___
Divide both sides of the inequality by 3.
​ 3x  ​ < ___
​  15  ​
3
3
Graph the solution on a number line. Use an open circle.
−1
0
1
2
3
4
5
x < 5
6
Lesson 2-A ~ Linear Inequalities 3
EXAMPLE 3
Solve the inequality and graph its solution on a number line.
5x + 1 ≤ 3x − 7
Solution
Subtract 3x from each side of the inequality.
5x + 1 ≤ 3x − 7
−3x
−3x
2x + 1 ≤ −7
Subtract 1 from each side.
−1 −1
2 x ≤ −8 2
2
Divide both sides by 2.
x ≤ −4
Graph the solution on a number line. Use a closed circle.
−5 −4 −3 −2 −1
0
1
2
One special rule applies to solving inequalities. Whenever you multiply or divide by a negative number
on both sides of the equation, you must flip the inequality symbol. For example, less than (<) would become
greater than (>) if you multiply or divide by a negative when performing inverse operations.
EXAMPLE 4
Solve the inequality −4x + 7 ≤ 19.
Solution
Subtract 7 from each side of the inequality.
−4x + 7 ≤ 19
−7 −7
−4x
____
Divide both sides by −4.
​    ​ ≤ ___
​ 12 ​ 
−4
−4
Since both sides were divided by a negative,
flip the inequality symbol. x ≥ −3
EXERCISES
Write an inequality for each graph shown. Use x as the variable.
1. −4 −3 −2 −1 0 1 2 3 2.
−3 −2 −1
0
1
2
3
4
4. −3 −2 −1
0
1
2
3
4
3. −3 −2 −1
4 0
1
Lesson 2-A ~ Linear Inequalities
2
3
4
Solve each inequality. Graph the solution on a number line.
5. 5x ≥ −20
6. x + 7 < 6
8. 2x + 7 > 15 9. __​  2x ​− 1 ≥ −4 14. −7 + 4x ≥ 3 − 6x 15. 2(x + 3) ≥ 5x + 12
7. 10 < 3x + 1
10. −3x − 4 < 5
11. 5(x + 3) ≤ 20 12. 7 > ___
​  x   ​+ 6 13. 9x < 2x − 35
−4
16. ​ _12 ​x + 8 < x + 4
17. A cargo elevator has a maximum carrying capacity of 240 pounds.
Each cargo box weighs 20 pounds.
a. Write and solve an inequality that represents the maximum
number of cargo boxes that the elevator can hold.
b. A 140-pound person rides in the elevator with the cargo boxes.
Write and solve an inequality that represents the maximum
number of cargo boxes the person can take with them in the
elevator.
18. Frankie has $400 in her bank account at the beginning of the summer. She wants to have at least $150 in
her account at the end of the summer. Each week she withdraws $22 for food and entertainment.
a. Write an inequality for this situation. Let x represent the number of weeks she withdraws money
from her account.
b. How many weeks can Frankie withdraw money from her account?
19. Ethan was at the beach. He wanted to spend $10 or less on a beach bike rental. The company he chose to
rent from charged an initial fee of $4 and an additional $0.35 per mile he rode.
a. Write an inequality for this situation. Let x represent the number of miles ridden.
b. How many miles can Ethan ride without going over his spending limit? Round to the nearest whole
mile.
20. Callie and BreShay went to the mall. They each spent the exact
same amount during the day. Callie spent less than or equal to $45.
BreShay spent more than $40. Create a number line that shows all
the possible amounts that Callie or BreShay could have spent
during the day.
Lesson 2-A ~ Linear Inequalities 5