Two-Step Inequalities

Grade 7 · algebra · 101 practice problems · read aloud

🔊 Listen to this explanation

Two-Step Inequalities

What Are Two-Step Inequalities? 🤔

An inequality, like 3x + 2 > 11, shows that two expressions are not equal. One side is greater than (>) or less than (<) the other. We solve them to find a range of possible numbers that make the inequality true, which is super useful for real-life situations like figuring out how many hours you need to work to buy a new video game.

How to Solve Them: Step-by-Step

  1. Step 1: Undo Addition/Subtraction. Use the inverse operation (the opposite) to isolate the term with the variable.
  2. Step 2: Undo Multiplication/Division. Use the inverse operation to completely isolate the variable.
  3. CRUCIAL CHECK: If you multiply or divide by a negative number, you MUST flip the inequality sign (e.g., > becomes <).

Visual Examples

Example 1: Solve 3x + 2 > 11

  • Step 1: Subtract 2 from both sides: 3x + 2 - 2 > 11 - 2 → 3x > 9
  • Step 2: Divide both sides by 3: 3x / 3 > 9 / 3 → x > 3

✅ The solution is all numbers greater than 3.

Example 2: Solve -2x - 4 ≤ 6

  • Step 1: Add 4 to both sides: -2x - 4 + 4 ≤ 6 + 4 → -2x ≤ 10
  • Step 2: Divide both sides by -2. FLIP THE SIGN! -2x / -2 ≥ 10 / -2 → x ≥ -5

✅ The solution is all numbers greater than or equal to -5.

Common Mistakes to Avoid 🚫

Forgetting to Flip the Inequality Sign: This is the #1 mistake! You only flip it when you multiply or divide by a negative number. Adding or subtracting a negative does NOT require flipping.

Incorrect Inverse Operations: Make sure you're doing the exact same thing to both sides of the inequality. Subtraction undoes addition; division undoes multiplication.

Tips & Tricks

Think "Balance": Imagine the inequality is a balanced scale. Whatever you do to one side, you must do to the other to keep it balanced.

Negative Number Reminder: Remember the phrase "Negative Flips" to help you remember when to flip the sign.

Check Your Answer: Pick a number from your solution set and plug it back into the original inequality to see if it makes a true statement.

How to Practice

  • Start with simple problems and gradually try ones with negative coefficients.
  • Write out every single step on paper—don't skip steps in your head!
  • Practice graphing your solutions on a number line to visualize the answer.
  • Ask your teacher for worksheets or find online practice quizzes.

Practice problems

6 of the 101, worked through step by step — try them before opening the answer.

1 3x - 7 > 14

Hint: First isolate the variable term by performing the inverse operation on the constant, then divide both sides by the coefficient. Remember to maintain the inequality direction unless dividing by a negative number.

Show the answer

Answer: x > 7

  1. Add 7 to both sides** We do this to isolate the term with the variable (3x) on one side. 3x − 7 + 7 > 14 + 7 3x > 21 **
  2. Divide both sides by 3** Since 3 is positive, the inequality sign stays the same. 3x / 3 > 21 / 3 x > 7 **Final answer:** x > 7

Let's solve the inequality step by step. We start with: 3x − 7 > 14 **

2 3x + 7 > 22

Hint: Isolate the variable by performing inverse operations on both sides of the inequality, remembering to maintain the direction of the inequality sign.

Show the answer

Answer: x > 5

  1. Subtract 7 from both sides to isolate the term with x. This is because we want to undo the addition of 7. 3x + 7 - 7 > 22 - 7 3x > 15
  2. Divide both sides by 3 to solve for x. Since 3 is positive, the inequality sign stays the same. 3x / 3 > 15 / 3 x > 5 So the solution is x > 5.

We start with the inequality: 3x + 7 > 22

3 3x - 7 ≥ 14

Hint: First isolate the variable term by performing the inverse operation on the constant, then divide by the coefficient, remembering to maintain the inequality direction unless dividing by a negative number.

Show the answer

Answer: x ≥ 7

  1. Add 7 to both sides to isolate the term with x. This is because we want to undo the subtraction of 7. 3x - 7 + 7 ≥ 14 + 7 3x ≥ 21
  2. Divide both sides by 3 to solve for x. Since 3 is positive, the inequality sign stays the same. 3x / 3 ≥ 21 / 3 x ≥ 7 So the solution is: x ≥ 7

We start with the inequality: 3x - 7 ≥ 14

4 2x - 7 > 15

Hint: First isolate the variable term by performing the inverse operation on the constant, then divide by the coefficient while remembering to maintain the inequality direction.

Show the answer

Answer: x > 11

  1. Add 7 to both sides. We do this to isolate the term with the variable (2x). 2x - 7 + 7 > 15 + 7 2x > 22
  2. Divide both sides by 2. We do this to solve for x. 2x / 2 > 22 / 2 x > 11 So the solution is: x > 11 This means any number greater than 11 makes the original inequality true.

Let's solve the inequality step by step. We start with: 2x - 7 > 15

5 3x + 7 ≤ 25

Hint: Isolate the variable by performing inverse operations on both sides of the inequality, remembering to maintain the direction of the inequality symbol.

Show the answer

Answer: x ≤ 6

  1. Subtract 7 from both sides of the inequality: 3x + 7 - 7 ≤ 25 - 7, which simplifies to 3x ≤ 18.
  2. Divide both sides by 3 to solve for x: 3x / 3 ≤ 18 / 3, which simplifies to x ≤ 6. The solution is x ≤ 6.

6 3x - 8 ≤ 19

Hint: Remember to perform the same operation on both sides of the inequality when solving, and pay attention to whether the inequality sign needs to be reversed.

Show the answer

Answer: x ≤ 9

  1. Start with the inequality 3x - 8 ≤ 19
  2. Add 8 to both sides to isolate the term with the variable: 3x - 8 + 8 ≤ 19 + 8
  3. Simplify both sides: 3x ≤ 27
  4. Divide both sides by 3 to solve for x: 3x ÷ 3 ≤ 27 ÷ 3
  5. Simplify: x ≤ 9 Since we divided by a positive number (3), the inequality sign does not change direction. The solution is x ≤ 9.
Practise this topic — 10 free problems, no signup →