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Uses of Two Step Inequalities

Sep 16, 2022

Key Concepts

• Solve two-step inequalities

• Solve more two-step inequalities

• Solve inequalities negative values

5.6 Solve Two-Step Inequalities  

  • To solve a two-step inequality, undo the addition or subtraction first, using inverse operations, and then undo the multiplication or division.  
  • The inverse operation of addition is subtraction and vice versa. 


Solve 2x + 1 < 7. 


First, we need to isolate the variable term on one side of the inequality.  

Here, on the left, 1 is added to the variable term, 2x.  

The inverse operation of addition is subtraction. So, subtract 1 from both sides. 

2x + 1 – 1 < 7 − 1         

2x < 6


Now, we have the variable x multiplied by 2. 

The inverse operation of multiplication is division. So, divide both sides by 2. 

2x /2 < 6 / 2


That is, the inequality is true for all values of x, which are less than 3. 

Therefore, the solutions to the inequality 2x+1<7 are all numbers less than 3. 

5.6.1. Solve Two-Step Inequalities  

Example 1: 

Tyler needs at least $205 for a new video game system. He already saved $30. He earns $7an hour at his job. Write and solve an inequality to find how many hours he will need to work to buy the system. 


30 + 7h ≥ 205 

30 – 30 + 7h ≥ 205 – 30 ( Subtract 30 from both sides.)

7h ≥ 175 (Divide both sides by 7.) 

h ≥ 25 

Tyler has to work 25 hours or more in order to make enough money. 

Example 2: 

Solve the inequality equation, then graph the solution. 

12 – 3a > 18 


12 – 3a > 18 

–12      –12     (Subtract 12 from both sides. ) 

−3a/−3 ≤ 6/3 (Simplify.) 

Divide both sides by –3. 

a < –2 

Graph the solution. 

5.6.2. Solve More Two-Step Inequalities  

Jerri has $17.50 and is going to the movie theater. A movie ticket costs $10, and snacks are $2.50 each. What maximum number of snacks can she buy? 


Cost of tickets and snacks ≤ 17. 50 

10 + 2.50 × number of snacks ≤17.50 

  10 + 2.50n ≤ 7.50 

10 – 10+2.50n ≤ 17.50 – 10 (Subtract 10 from both sides.)  

2.50n ≤ 7.50 

2.50n/2.50 ≤ 7.50/ 2.50 (Divide both sides by 2.50.) 

n ≤ 3 

Jerri can buy a maximum of 3 snacks. 

5.6.3. Solve Inequalities Negative Values 


Solve the inequality –2x + 14 ≥ –18, then graph the solution.  


–2x + 14 ≥ –18  

–2x + 14-14 ≥ –18 – 14  ……(Subtract 14 from both sides.)  

–2x ≥ –32 ……(Simplify the inequality.) 

−2x/−2 ≥ −32/−2……(Divide both sides by –2) 

x ≤ 16 


  1. Which graph shows the solution to 2x -10 ≥ 4?

2. Solve the inequality equations. -9x+ 4 ≤31

3. 18n- 22 ≤ 32

4. 3x -11 ≥ -20

5. -3x + 31 < 25

6. Triniti had $500 in a savings account at the beginning of the summer. She wants to have at least $200 in the account by the end of the summer. She withdraws $2S each week for food, clothes, and movie tickets. a. Write an inequality that represents Triniti’s situation. b. How many weeks can Triniti withdraw money from her account Justify your answer. c. Graph the solution on a number line.

7. Kevin has $25. MP3 downloads cost $0.75 each. How many songs can he download and still have $13 left to spend? a. Write an inequality that represents Kevin’s situation. b. How many MP3s can Kevin purchase? Justify your answer. c. Graph the solution on a number line.

8. 5x+2 ≥ 12

9. A taxi company charges a flat fee of $3.50 per ride plus an additional $0.65 per mile. If Ian has only $10, what is the farthest he can ride?

 10. Nadia has a daily budget of $94 for a car rental. Write and solve an inequality to find the greatest distance Nadia can drive each day while staying within her budget Oar Pats! $$0 Pat Day Plus $020 per mile

Concept Map

What have we learned:

• Write two step inequalities

• Solve two-step inequalities

• Solve more two-step inequalities

• Solve inequalities negative values

• Graph the solution


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