In this chapter, we’ll learn how to solve inequalities just like we solve simple equations. But there’s one key difference: we’re not looking for just one value—we’re looking for a range of values that make the inequality true.
To solve an inequality means to find all possible values that make the inequality true.
🧠 Think of it like solving an equation:
For an equation: x + 3 = 10, we find x = 7.
For an inequality: x + 3 < 10, we find all x values that are less than 7.
✅ Final answer: x < 7
Let’s look at how to solve inequalities by adding or subtracting from both sides—just like equations.
Example 1: Solve x + 5 < 12
Subtract 5 from both sides:
x + 5 - 5 < 12 - 5
x < 7
✅ Answer: x < 7
✏️ Graph:
Open circle at 7
Arrow to the left
Example 2: Solve x - 4 ≥ 9
Add 4 to both sides:
x - 4 + 4 ≥ 9 + 4
x ≥ 13
✅ Answer: x ≥ 13
✏️ Graph:
Closed circle at 13
Arrow to the right
Solving inequalities using multiplication or division is just like solving equations—with one special rule.
🛑 IMPORTANT RULE:
When you multiply or divide both sides of an inequality by a negative number, you must flip the inequality sign!
Example 3: Solve 5x > 20
Divide both sides by 5:
x > 4
✅ Final answer: x > 4
Example 4: Solve -3x ≤ 12
Divide both sides by -3
Flip the inequality sign!
x ≥ -4
✅ Final answer: x ≥ -4
✏️ Remember: Flipping the sign only happens with negative multiplication or division.
Solve each inequality and graph the solution:
x + 6 ≤ 10
x - 2 > 3
4x ≥ 20
-2x < 6
x ÷ 5 ≤ 3
Problem 1:
You need at least 75 points to pass a test. You already have 63 points. How many more points (𝑥) do you need?
Inequality:
x + 63 ≥ 75
Solve:
x ≥ 12
✅ You need at least 12 more points.
Problem 2:
A water bottle holds less than 2 liters. If it currently has 0.8 liters, how much more can it hold?
Let x be the amount of water we can still add:
x + 0.8 < 2
Solve:
x < 1.2
✅ We can add less than 1.2 liters more.
Step Rule
1 Isolate the variable (get it by itself on one side) 2 Use inverse operations (subtract, add, multiply, or divide) 3 If you multiply or divide by a negative, flip the inequality sign 4 Graph your final answer on a number line
Solving inequalities is just like solving equations—but instead of =, we use inequality symbols.
Use inverse operations to isolate the variable.
Always flip the inequality sign when multiplying or dividing by a negative number.
After solving, graph your answer to show all possible solutions.
Next, we’ll take it a step further and solve two-step inequalities—those that involve both addition/subtraction and multiplication/division in the same problem!