Practice solving and interpreting inequalities. Inequalities use symbols such as <, >, ≤, and ≥ to compare values instead of showing that two sides are exactly equal.
Inequality symbols compare two values. The symbol points toward the smaller value and opens toward the larger value. Understanding the symbol is the first step before solving inequalities.
Problem: What does x > 5 mean?
Step 1: Read the symbol.
The symbol > means “greater than.”
Step 2: Interpret the statement.
x > 5 means x can be any number greater than 5.
Examples: 6, 7, 10, and 100 are all possible solutions.
Answer: x is greater than 5.
You are ready to move on when you can read each inequality symbol and explain what values are allowed.
Solving a one-step inequality is similar to solving a one-step equation. Use the opposite operation to isolate the variable. The inequality symbol usually stays the same when you add or subtract the same number on both sides.
Problem: Solve x + 4 < 10
Step 1: The variable x has 4 added to it.
Step 2: Use the opposite operation. Subtract 4 from both sides.
x + 4 - 4 < 10 - 4
Step 3: Simplify.
x < 6
Answer: x < 6
You are ready to move on when you can isolate the variable using one opposite operation and keep the inequality statement true.
When solving inequalities, there is one special rule: if you multiply or divide both sides by a negative number, you must flip the inequality symbol. This keeps the inequality statement true.
Problem: Solve -2x < 10
Step 1: The variable x is multiplied by -2.
Step 2: Divide both sides by -2.
-2x ÷ -2 < 10 ÷ -2
Step 3: Since we divided by a negative number, flip the inequality symbol.
x > -5
Answer: x > -5
You are ready to move on when you remember to flip the inequality symbol every time you multiply or divide by a negative number.
A two-step inequality is solved almost like a two-step equation. First, undo addition or subtraction. Then undo multiplication or division. Remember: if you multiply or divide by a negative number, flip the inequality symbol.
Problem: Solve 3x + 2 < 14
Step 1: Subtract 2 from both sides.
3x + 2 - 2 < 14 - 2
3x < 12
Step 2: Divide both sides by 3.
3x ÷ 3 < 12 ÷ 3
x < 4
Answer: x < 4
You are ready to move on when you can solve a two-step inequality and know when the inequality symbol should stay the same or flip.
Checking an inequality solution means substituting a possible value into the original inequality. If the statement is true, the value is a solution. If the statement is false, the value is not a solution.
Problem: Is x = 2 a solution to x < 4?
Step 1: Start with the original inequality.
x < 4
Step 2: Substitute x = 2.
2 < 4
Step 3: Decide if the statement is true.
2 is less than 4, so the statement is true.
Answer: Yes, x = 2 is a solution.
You are ready to move on when you can substitute a value into an inequality and decide whether the final statement is true or false.