Graph Inequalities

Grade 7 · algebra · 91 practice problems · read aloud

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Graphing Inequalities 🌈

What is it and Why is it Useful?

Graphing an inequality shows all the possible solutions to an inequality on a number line. Instead of just one answer, you get a whole range! This is super useful for representing real-world situations like "I need to save more than $20" or "The temperature was less than or equal to 5°C."

How to Graph an Inequality: Step-by-Step

  1. Solve the inequality for the variable (get x by itself), just like an equation.
  2. Find your number on the number line.
  3. Choose the right circle:
    • Use an OPEN circle (○) for < (less than) or > (greater than).
    • Use a CLOSED circle (●) for ≤ (less than or equal to) or ≥ (greater than or equal to).
  4. Shade the number line in the direction of all the possible solutions.
    • Shade LEFT for < or ≤.
    • Shade RIGHT for > or ≥.

Visual Examples

Example 1: x > 3

Step 1: The inequality is already solved. Our number is 3.
Step 2: Since it's ">", we use an OPEN circle on 3.
Step 3: "Greater than" means we shade to the RIGHT.
Visual: Number line with open circle on 3, arrow shading to the right.

Example 2: y ≤ -1

Step 1: The inequality is solved. Our number is -1.
Step 2: Since it's "≤", we use a CLOSED circle on -1.
Step 3: "Less than or equal to" means we shade to the LEFT.
Visual: Number line with closed circle on -1, arrow shading to the left.

Common Mistakes to Avoid 🚫

Using the wrong circle: This is the #1 mistake! Always check your symbol. An open circle means the number itself is not a solution. A closed circle means it is a solution.

Shading the wrong way: Remember, smaller numbers are on the left, and larger numbers are on the right. If x is greater than a number, the solutions are bigger, so shade right!

Tips & Tricks

The Hungry Alligator 🐊: Imagine the inequality symbol is a hungry alligator's mouth. It always opens towards the bigger number (or variable side). This can help you remember which way to shade!

Circle Connection: The closed circle for "≤" and "≥" looks like it has a line under it, just like the inequality symbol has a line under it for "or equal to."

How to Practice

  • Start simple: Practice graphing inequalities like x < 5 and n ≥ 0.
  • Mix it up: Create a worksheet with a variety of symbols and numbers (positive, negative, fractions).
  • Write your own: Think of a real-life rule (e.g., "You must be at least 13 to sign up") and write and graph the inequality.
  • Check your work: Use an online graphing tool to see if your number line matches!

Practice problems

6 of the 91, worked through step by step — try them before opening the answer.

1 2x + 7 > 15

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 > 4

  1. Subtract 7 from both sides to isolate the term with x. 2x + 7 - 7 > 15 - 7 2x > 8
  2. Divide both sides by 2 to solve for x. 2x / 2 > 8 / 2 x > 4 Final answer: x > 4 Explanation: Since we only added or subtracted numbers and divided by a positive number, the direction of the inequality sign remains the same.

We start with the inequality: 2x + 7 > 15

2 3x + 7 > 16

Hint: Isolate the variable by performing inverse operations on both sides of the inequality, then check if the inequality sign direction needs to be maintained.

Show the answer

Answer: x > 3

  1. Subtract 7 from both sides to isolate the term with x. Reason: We want to get the x-term alone on one side. Subtracting 7 from both sides keeps the inequality balanced. 3x + 7 - 7 > 16 - 7 This simplifies to: 3x > 9
  2. Divide both sides by 3 to solve for x. Reason: Since 3 is multiplied by x, dividing both sides by 3 will isolate x. We do not flip the inequality sign because we are dividing by a positive number. 3x / 3 > 9 / 3 This simplifies to: x > 3 Final answer: x > 3

We start with the inequality: 3x + 7 > 16

3 3(2x - 5) ≤ 4x + 7

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

Show the answer

Answer: x ≤ 11

  1. Distribute the 3** Multiply 3 by each term inside the parentheses: 3 * 2x = 6x 3 * (-5) = -15 So we get: 6x - 15 ≤ 4x + 7 **
  2. Move the x terms to one side** Subtract 4x from both sides: 6x - 4x - 15 ≤ 4x - 4x + 7 2x - 15 ≤ 7 **
  3. Move constant terms to the other side** Add 15 to both sides: 2x - 15 + 15 ≤ 7 + 15 2x ≤ 22 **
  4. Solve for x** Divide both sides by 2: 2x / 2 ≤ 22 / 2 x ≤ 11 **Final answer:** x ≤ 11

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

4 2(3x - 5) ≥ 4x + 8

Hint: Distribute the coefficient outside the parentheses first, then combine like terms by moving all variable terms to one side and constants to the other. Remember to reverse the inequality sign if you multiply or divide by a negative number.

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. Get all x terms on one side** Subtract 4x from both sides: 6x - 4x - 10 ≥ 8 2x - 10 ≥ 8 **
  3. Isolate the x term** Add 10 to both sides: 2x - 10 + 10 ≥ 8 + 10 2x ≥ 18 **
  4. Solve for x** Divide both sides by 2: x ≥ 9 **
  5. Interpret the result** The solution is x ≥ 9, which means all numbers greater than or equal to 9 satisfy the original inequality. **Final answer:** x ≥ 9

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

5 2(3x - 5) ≥ 4x + 6

Hint: Distribute first, then combine like terms on each side before isolating the variable. Remember to reverse the inequality sign if you multiply or divide by a negative number.

Show the answer

Answer: x ≥ 8

  1. Distribute the 2 on the left side** 2 * 3x = 6x 2 * (-5) = -10 So we have: 6x - 10 ≥ 4x + 6 **
  2. Get all x terms on one side** Subtract 4x from both sides: 6x - 4x - 10 ≥ 4x - 4x + 6 2x - 10 ≥ 6 **
  3. Isolate the x term** Add 10 to both sides: 2x - 10 + 10 ≥ 6 + 10 2x ≥ 16 **
  4. Solve for x** Divide both sides by 2: x ≥ 8 **Final answer:** x ≥ 8

Let's solve the inequality step by step. We start with: 2(3x - 5) ≥ 4x + 6 **

6 3(2x - 5) ≥ 4x + 7

Hint: First distribute any multiplication across parentheses, then combine like terms on each side before isolating the variable.

Show the answer

Answer: x ≥ 11

  1. Distribute the 3** Multiply 3 by each term inside the parentheses: 3 * 2x = 6x 3 * (-5) = -15 So we get: 6x - 15 ≥ 4x + 7 **
  2. Move all terms with x to one side** Subtract 4x from both sides: 6x - 4x - 15 ≥ 4x - 4x + 7 2x - 15 ≥ 7 **
  3. Move constant terms to the other side** Add 15 to both sides: 2x - 15 + 15 ≥ 7 + 15 2x ≥ 22 **
  4. Solve for x** Divide both sides by 2: x ≥ 11 **
  5. Interpret the result** The solution means x can be any number greater than or equal to 11. **Final answer:** x ≥ 11

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

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