Linear Inequalities

Grade 9 · algebra · 102 practice problems · read aloud

🔊 Listen to this explanation

Linear Inequalities: Your Guide to Comparing Values

What is a Linear Inequality?

A linear inequality is like a linear equation, but instead of an equals sign (=), it uses inequality symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). It shows a range of possible solutions, not just one answer. We use them to describe real-world situations like "spending less than $50" or "scoring at least 70 points to pass."

How to Solve Linear Inequalities

You solve them much like linear equations, with one crucial difference.

  1. Simplify: Use the distributive property and combine like terms on each side.
  2. Isolate the Variable: Use inverse operations to get the variable term by itself.
    • ➕➖ Adding or subtracting a number from both sides is always safe.
    • ✖️➗ Multiplying or dividing both sides by a negative number flips the inequality symbol. This is the key rule!
  3. Check Your Solution: Pick a number from your solution set and plug it back into the original inequality to verify it's true.

Worked Examples

Example 1: No Flipping Needed

Solve: \( 2x - 5 ≤ 7 \)

  1. Add 5 to both sides: \( 2x ≤ 12 \)
  2. Divide both sides by 2 (a positive number): \( x ≤ 6 \)

Solution: \( x ≤ 6 \)

Example 2: Flip the Symbol! 🚨

Solve: \( -3x + 6 > 15 \)

  1. Subtract 6 from both sides: \( -3x > 9 \)
  2. Divide both sides by -3 (a negative number): \( x < -3 \) (The > flips to <)

Solution: \( x < -3 \)

Common Mistakes to Avoid

  • Forgetting to Flip the Inequality: This is the #1 error. Always check if you are multiplying or dividing by a negative.
  • Incorrectly Graphing the Solution: For \( x ≤ 6 \), use a closed circle at 6 and shade left. For \( x < -3 \), use an open circle at -3 and shade left.
  • Treating ≤ and < as the Same: The "or equal to" part matters! It determines if the endpoint is included in the solution.

Tips & Tricks

  • The "L" Method: The symbols < and > can look like an "L". The "L" stands for "Less than." No "L" means "Greater than."
  • Negative Flipping Memory Aid: Imagine the inequality symbol is an alligator's mouth. It always "eats" the bigger number. If you multiply/divide by a negative, it gives the alligator indigestion, so it flips its mouth the other way!
  • Check with a Number: After solving, test a number from your solution set in the original problem to be sure.

How to Practice

Mastery comes with consistent practice!

  • Start with simple one-step inequalities (e.g., \( x + 3 > 1 \), \( -2y ≤ 8 \)).
  • Move on to multi-step problems involving distributing and combining like terms.
  • Always graph your solutions on a number line to build a visual connection.
  • Create your own word problems. For example, "If a pizza costs $12 and I have $50, how many pizzas can I buy?" ( \( 12x ≤ 50 \) ).

Practice problems

6 of the 102, worked through step by step — try them before opening the answer.

1 2|x - 3| - 5 > 7

Hint: Remember that absolute value inequalities require considering both positive and negative cases. Isolate the absolute value expression first, then create two separate inequalities.

Show the answer

Answer: x < -3 or x > 9

  1. Add 5 to both sides: 2|x - 3| > 12
  2. Divide both sides by 2: |x - 3| > 6
  3. For absolute value greater than a positive number, create two inequalities: x - 3 > 6 OR x - 3 < -6
  4. Solve the first inequality: x > 9
  5. Solve the second inequality: x < -3
  6. Combine the solutions: x < -3 or x > 9

2 2|x - 3| - 5 ≥ 7

Hint: Remember that absolute value inequalities require considering both positive and negative cases. Isolate the absolute value expression first.

Show the answer

Answer: x ≤ -3 or x ≥ 9

  1. Add 5 to both sides: 2|x - 3| - 5 + 5 ≥ 7 + 5 → 2|x - 3| ≥ 12
  2. Divide both sides by 2: |x - 3| ≥ 6
  3. For absolute value inequalities with ≥, we create two inequalities: x - 3 ≤ -6 OR x - 3 ≥ 6
  4. Solve x - 3 ≤ -6: x ≤ -6 + 3 → x ≤ -3
  5. Solve x - 3 ≥ 6: x ≥ 6 + 3 → x ≥ 9
  6. Combine solutions: x ≤ -3 OR x ≥ 9

3 3|2x - 5| - 7 < 8

Hint: Remember that absolute value inequalities require considering both positive and negative cases. Isolate the absolute value expression first, then create two separate inequalities.

Show the answer

Answer: -1 < x < 5

  1. Add 7 to both sides: 3|2x - 5| < 15
  2. Divide both sides by 3: |2x - 5| < 5
  3. Write the compound inequality: -5 < 2x - 5 < 5
  4. Add 5 to all parts: 0 < 2x < 10
  5. Divide all parts by 2: 0 < x < 5
  6. The solution is -1 < x < 5 Final answer: -1 < x < 5

4 2(3x - 5) ≤ 4x + 8

Hint: Distribute the coefficient outside the parentheses first, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x ≤ 9

  1. Distribute the 2 on the left side** 2 * 3x = 6x 2 * (-5) = -10 So we have: 6x - 10 ≤ 4x + 8 **
  2. Move the x terms to one side** Subtract 4x from both sides: 6x - 4x - 10 ≤ 8 2x - 10 ≤ 8 **
  3. Move constant terms to the other side** Add 10 to both sides: 2x - 10 + 10 ≤ 8 + 10 2x ≤ 18 **
  4. Divide by 2** x ≤ 18 / 2 x ≤ 9 **Final answer:** x ≤ 9

Let's solve the inequality step by step. We start with: 2(3x - 5) ≤ 4x + 8 **

5 2(3x - 5) + 4 > 7x - 1

Hint: When solving inequalities, remember to distribute any coefficients outside parentheses first, then combine like terms on each side before isolating the variable. Pay attention to how the inequality sign behaves when multiplying or dividing by negative numbers.

Show the answer

Answer: x > 5

Linear inequalities follow similar solving steps as equations, but with special attention to inequality direction. When you multiply or divide both sides by a negative number, the inequality sign reverses direction. The solution represents all values that make the inequality true, often expressed as an inequality statement or on a number line.

6 3(2x - 5) + 7 < 4x - 2

Hint: Distribute any coefficients outside parentheses first, then combine like terms on each side before isolating the variable term.

Show the answer

Answer: x < 3

  1. Distribute the 3** 3 * 2x = 6x 3 * (-5) = -15 So we have: 6x - 15 + 7 < 4x - 2 **
  2. Combine like terms on the left** -15 + 7 = -8 So: 6x - 8 < 4x - 2 **
  3. Get all x terms on one side** Subtract 4x from both sides: 6x - 8 - 4x < 4x - 2 - 4x 2x - 8 < -2 **
  4. Isolate the x term** Add 8 to both sides: 2x - 8 + 8 < -2 + 8 2x < 6 **
  5. Solve for x** Divide both sides by 2: x < 3 **Final answer:** x < 3

Let's solve the inequality step-by-step. We start with: 3(2x - 5) + 7 < 4x - 2 **

Practise this topic — 10 free problems, no signup →