Linear Inequalities

Grade 8 · algebra · 99 practice problems · read aloud

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Linear Inequalities: The Basics

A linear inequality is like a linear equation (e.g., 2x + 3 = 7) 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). We use them to show a range of possible solutions, not just one answer. They are useful for real-world situations like budgeting (spending less than you earn) or setting speed limits.

How to Solve Linear Inequalities

  1. Isolate the Variable: Use inverse operations (add, subtract, multiply, divide) to get the variable by itself on one side of the inequality, just like you would with an equation.
  2. CRITICAL RULE: If you multiply or divide both sides by a negative number, you MUST flip the inequality symbol. ↕️
  3. Graph the Solution: On a number line, use an open circle for < or > and a closed circle for ≤ or ≥. Then shade in the direction of all the possible solutions.

Worked Examples

Example 1: No Flipping Needed

Solve and graph: x - 5 ≤ 2

  1. Add 5 to both sides: x ≤ 7

Graph: A closed circle on 7, with shading to the left. ✅

Example 2: Flip the Symbol! 🚨

Solve and graph: -3x > 12

  1. Divide both sides by -3.
  2. FLIP THE SYMBOL from > to <: x < -4

Graph: An open circle on -4, with shading to the left.

Common Mistakes to Avoid

Forgetting to Flip the Inequality: This is the #1 mistake! You only flip the sign when multiplying or dividing by a negative number. Adding or subtracting a negative does NOT require flipping.

Incorrect Graphing: Using an open circle when you should use a closed circle (or vice-versa). Remember: ≤ and ≥ get a closed circle because they include the number.

Tips & Tricks

The "Hungry Alligator" 🐊: Imagine the inequality symbol is a hungry alligator's mouth. It always opens to eat the larger number! This can help you remember which way the symbol points.

Check Your Answer: Pick a number from your shaded solution and plug it back into the original inequality. If it makes a true statement, you're correct!

How to Practice

  • Start with simple one-step inequalities before moving to multi-step problems.
  • Practice graphing your solutions on a number line every single time.
  • Create your own word problems. For example, "I have $20 and want to buy snacks that cost $3 each. Write and solve an inequality to find how many I can buy."

Practice problems

6 of the 99, worked through step by step — try them before opening the answer.

1 2x + 3y ≤ 12; x ≥ 0; y ≥ 0

Hint: To find the boundary points, set each variable to zero and solve for the other variable in the inequality.

Show the answer

Answer: x ≤ 6; y ≤ 4

  1. Find the maximum possible value of x** To find the largest possible x, set y to its smallest possible value, which is y = 0 (since y ≥ 0). Substitute y = 0 into the inequality: 2x + 3(0) ≤ 12 2x ≤ 12 Divide both sides by 2: x ≤ 6 So, x can be at most 6. --- **
  2. Find the maximum possible value of y** To find the largest possible y, set x to its smallest possible value, which is x = 0 (since x ≥ 0). Substitute x = 0 into the inequality: 2(0) + 3y ≤ 12 3y ≤ 12 Divide both sides by 3: y ≤ 4 So, y can be at most 4. --- **
  3. Interpret the result** From the above: - When y = 0, x ≤ 6 - When x = 0, y ≤ 4 But note: x and y can be smaller than these maximums, and they can vary together as long as 2x + 3y ≤ 12. However, the problem's correct answer is given as x ≤ 6 and y ≤ 4, which are the upper bounds for x and y individually under the given constraints. --- **Final Answer:** x ≤ 6 y ≤ 4

Let's solve the inequality step by step. We are given: 2x + 3y ≤ 12 x ≥ 0 y ≥ 0 --- **

2 2x + 3y ≤ 12 where x = 2, y = 3

  1. A) false
  2. B) true

Hint: Substitute the given values for the variables into the inequality and check if the resulting statement is correct.

Show the answer

Answer: B) true

  1. Substitute the given values of x and y into the expression 2x + 3y. 2x + 3y = 2*(2) + 3*(3)
  2. Perform the multiplication. 2*(2) = 4 3*(3) = 9
  3. Add the results together. 4 + 9 = 13
  4. Compare the result to 12 using the inequality. We have 13 ≤ 12.
  5. Determine if this statement is true or false. 13 is not less than or equal to 12. It is greater than 12. So the statement 13 ≤ 12 is false.
  6. Interpret the result. Since substituting x=2 and y=3 gives a false statement, the inequality 2x + 3y ≤ 12 is false for the point (2,3).

Let's check if the inequality 2x + 3y ≤ 12 is true when x = 2 and y = 3. The correct answer is: false

3 2x + 3y ≤ 12 where x = 2 and y = 3

  1. A) true
  2. B) false

Hint: Substitute the given values for the variables into the inequality and check if the resulting statement is correct.

Show the answer

Answer: A) true

  1. Write down the inequality and the given values. Inequality: 2x + 3y ≤ 12 Given: x = 2, y = 3
  2. Substitute the values of x and y into the expression on the left side. 2x + 3y becomes 2*(2) + 3*(3)
  3. Perform the multiplication first (following order of operations). 2*(2) = 4 3*(3) = 9
  4. Add the results together. 4 + 9 = 13
  5. Compare the result (13) to the number on the right side of the inequality (12). We check: Is 13 ≤ 12?
  6. Analyze the comparison. 13 is not less than or equal to 12. 13 is greater than 12. Therefore, the statement 13 ≤ 12 is false.
  7. Final conclusion. Since substituting the values makes the inequality false,

Let's check if the inequality is true when x = 2 and y = 3. the correct answer is false. ANSWER: false

4 Is (5, 0) a solution to y ≤ 2x - 5?

Hint: Substitute the x-coordinate and y-coordinate into the inequality. Then check if the resulting statement is true or false.

Show the answer

Answer: Yes

  1. Identify the coordinates. x = 5, y = 0.
  2. Substitute into the inequality y ≤ 2x - 5. 0 ≤ 2(5) - 5
  3. Simplify the right side. 0 ≤ 10 - 5 0 ≤ 5
  4. Check if the statement is true. 0 is less than or equal to 5, so the statement is true. The point (5, 0) satisfies the inequality.

The answer is Yes.

5 Is (3, 7) a solution to y > 3x - 5?

Hint: Substitute the x and y coordinates into the inequality and check if the resulting statement is true.

Show the answer

Answer: No

  1. Substitute x = 3 and y = 7 into the inequality y > 3x - 5.
  2. Left side: y = 7. Right side: 3(3) - 5 = 9 - 5 = 4.
  3. Check the inequality: 7 > 4. This is true.
  4. Since 7 is greater than 4, the point (3, 7) satisfies the inequality.

The answer is Yes.

6 Is (3, 5) a solution to y > 3x - 7?

Hint: Substitute the x and y values from the point into the inequality and check if the resulting statement is true.

Show the answer

Answer: No

  1. Substitute x = 3 and y = 5 into the inequality y > 3x - 7.
  2. Left side: y = 5.
  3. Right side: 3(3) - 7 = 9 - 7 = 2.
  4. Check the inequality: 5 > 2.
  5. Since 5 is greater than 2, the inequality is true. Therefore, (3, 5) is a solution to y > 3x - 7.

The answer is Yes.

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