In this chapter, we’ll learn how to solve two-step inequalities, which require two operations to isolate the variable. Just like with equations, we’ll work step-by-step to undo the operations, and we’ll still follow the rule about flipping the inequality sign when needed.
A two-step inequality is an inequality that requires two inverse operations (such as subtraction and division) to get the variable by itself.
📘 Example:
3x + 5 < 20
We need to:
Subtract 5
Then divide by 3
Just like two-step equations—but remember, the solution is a range of values.
Let’s go through the process using clear examples.
Example 1: Solve 2x + 3 ≤ 11
Step 1: Subtract 3 from both sides
2x + 3 - 3 ≤ 11 - 3
2x ≤ 8
Step 2: Divide both sides by 2
x ≤ 4
✅ Final Answer: x ≤ 4
✏️ Graph: Closed circle at 4, arrow to the left
Example 2: Solve 5x - 2 > 13
Step 1: Add 2 to both sides
5x - 2 + 2 > 13 + 2
5x > 15
Step 2: Divide by 5
x > 3
✅ Final Answer: x > 3
✏️ Graph: Open circle at 3, arrow to the right
🛑 Remember:
If you multiply or divide both sides by a negative number, you must flip the inequality sign.
Example 3: Solve -4x + 6 ≥ -10
Step 1: Subtract 6
-4x ≥ -16
Step 2: Divide by -4 → flip the sign
x ≤ 4
✅ Final Answer: x ≤ 4
✏️ Graph: Closed circle at 4, arrow to the left
Problem:
A taxi ride costs $4 for the base fare plus $2 per mile. You want to spend no more than $18.
Let 𝑥 = number of miles
Inequality: 2x + 4 ≤ 18
Step 1: Subtract 4
2x ≤ 14
Step 2: Divide by 2
x ≤ 7
✅ You can ride up to 7 miles.
Solve and graph each inequality:
3x + 2 < 17
6x - 5 ≥ 19
-2x + 4 < 10
-5x - 1 ≥ 9
7x - 14 ≤ 0
Problem 1:
A carnival charges $6 for admission and $2 per game. You have at most $20 to spend. How many games can you play?
Let 𝑥 = number of games
Inequality: 6 + 2x ≤ 20
✅ Solve: x ≤ 7
Problem 2:
You’re saving for a video game that costs $60. You already saved $20, and you earn $8 per week. After how many weeks will you have at least $60?
Let 𝑤 = weeks
Inequality: 8w + 20 ≥ 60
✅ Solve: w ≥ 5
✅ Key Takeaways:
Solve two-step inequalities using inverse operations, just like two-step equations.
Use the same order: undo addition/subtraction, then multiplication/division.
Flip the inequality if you divide or multiply by a negative number.
Always graph your solution and label it clearly.
✅ Coming Up in Chapter 5:
In the next chapter, we’ll explore compound inequalities, where two inequalities are combined into one expression using the words "and" or "or".