Graph Inequalities

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Answer Key & Explanations

Graph Inequalities · Grade 7 · Worksheet 1

1. Emma is organizing a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $75 plus $8 per shirt. Emma's budget for t-shirts is at most $500. Write an inequality to represent the situation, then determine the maximum number of t-shirts she can order without exceeding her budget. Answer: 53 Solution: Let x represent the number of t-shirts Emma can order. The total cost is the setup fee plus the cost per shirt: 75 + 8x The budget constraint means the total cost must be less than or equal to $500: 75 + 8x ≤ 500 Subtract 75 from both sides: 8x ≤ 425 Divide both sides by 8: x ≤ 53.125 Since Emma…
   Full step-by-step solution

   Step 1: Let x represent the number of t-shirts Emma can order. Step 2: The total cost is the setup fee plus the cost per shirt: 75 + 8x Step 3: The budget constraint means the total cost must be less than or equal to $500: 75 + 8x ≤ 500 Step 4: Subtract 75 from both sides: 8x ≤ 425 Step 5: Divide both sides by 8: x ≤ 53.125 Step 6: Since Emma cannot order a fraction of a shirt, the maximum number of whole shirts is 53. The answer is 53.

2. Mere is buying supplies for a school project. Each package of markers costs $8. Mere has a budget of $50 to spend on markers. What is the maximum number of packages Mere can buy? Graph the solution on a number line. Answer: 6 Solution: Let p = number of packages. Inequality: 8p ≤ 50. Solve: p ≤ 50 / 8 = 6.
   Full step-by-step solution

   Step 1: Let p = number of packages. Inequality: 8p ≤ 50. Step 2: Solve: p ≤ 50 / 8 = 6. Step 3: Since p must be a whole number, the maximum is 6. Step 4: Graph: closed circle at 6, arrow to the left.

3. Aroha is graphing the solution to the inequality 3x - 5 > 7 on a number line. First, solve the inequality. Then, describe what the graph will look like: What type of circle (open or closed) will be at the boundary point, and in which direction (left or right) will the ray be shaded? Answer: Open circle at 4, shaded to the right Solution: Solve the inequality 3x - 5 > 7. Add 5 to both sides: 3x - 5 + 5 > 7 + 5, which simplifies to 3x > 12. Divide both sides by 3: 3x / 3 > 12 / 3, which gives x > 4.
   Full step-by-step solution

   Step 1: Solve the inequality 3x - 5 > 7. Add 5 to both sides: 3x - 5 + 5 > 7 + 5, which simplifies to 3x > 12. Divide both sides by 3: 3x / 3 > 12 / 3, which gives x > 4. Step 2: Interpret the solution. x > 4 means all numbers greater than 4, but not including 4 itself. Step 3: Determine the circle type. Since the inequality is strictly greater than (not greater than or equal to), we use an open circle at 4. Step 4: Determine the shading direction. Since x is greater than 4, the ray is shaded to the right. Final answer: Open circle at 4, shaded to the right.

4. Graph 4(2x - 6) ≥ 40 Answer: x ≥ 8 Solution: Distribute the 4: 4(2x - 6) = 8x - 24, so the inequality becomes 8x - 24 ≥ 40. Add 24 to both sides: 8x - 24 + 24 ≥ 40 + 24 → 8x ≥ 64. Divide both sides by 8: 8x/8 ≥ 64/8 → x ≥ 8.
   Full step-by-step solution

   Step 1: Distribute the 4: 4(2x - 6) = 8x - 24, so the inequality becomes 8x - 24 ≥ 40. Step 2: Add 24 to both sides: 8x - 24 + 24 ≥ 40 + 24 → 8x ≥ 64. Step 3: Divide both sides by 8: 8x/8 ≥ 64/8 → x ≥ 8. Step 4: Graph on a number line: Draw a closed circle at 8 (since the inequality includes 'equal to'), and shade the number line to the right of 8 (since x is greater than or equal to 8). The solution is x ≥ 8.

5. Emma is organizing a school fundraiser and needs to order t-shirts. The printing company charges a $50 setup fee plus $8 per shirt. Emma's budget for t-shirts is $500. Write an inequality to represent the maximum number of shirts she can order, then solve for the actual maximum number of whole shirts she can purchase. Answer: 56 Solution: Let x represent the number of shirts. The total cost is the setup fee plus cost per shirt: 50 + 8x Set up the inequality for staying within budget: 50 + 8x ≤ 500 Subtract 50 from both sides: 8x ≤ 450 Divide both sides by 8: x ≤ 56.25 Since we can only order whole shirts, the maximum number is 56.
   Full step-by-step solution

   Step 1: Let x represent the number of shirts. Step 2: The total cost is the setup fee plus cost per shirt: 50 + 8x Step 3: Set up the inequality for staying within budget: 50 + 8x ≤ 500 Step 4: Subtract 50 from both sides: 8x ≤ 450 Step 5: Divide both sides by 8: x ≤ 56.25 Step 6: Since we can only order whole shirts, the maximum number is 56. The answer is 56 shirts.

6. Graph 3(2x - 9) + 5 > 4x - 11 Answer: x > 5.5 Solution: Distribute the 3: 3(2x - 9) + 5 > 4x - 11 becomes 6x - 27 + 5 > 4x - 11. Combine like terms on the left: 6x - 22 > 4x - 11. Subtract 4x from both sides: 6x - 4x - 22 > -11, which gives 2x - 22 > -11.
   Full step-by-step solution

   Step 1: Distribute the 3: 3(2x - 9) + 5 > 4x - 11 becomes 6x - 27 + 5 > 4x - 11. Step 2: Combine like terms on the left: 6x - 22 > 4x - 11. Step 3: Subtract 4x from both sides: 6x - 4x - 22 > -11, which gives 2x - 22 > -11. Step 4: Add 22 to both sides: 2x > 11. Step 5: Divide both sides by 2: x > 5.5. The solution is x > 5.5. On a number line, draw an open circle at 5.5 and shade to the right.

7. 2(3x - 7) ≤ 5x + 4 Answer: x ≤ 18 Solution: Distribute the 2: 2 × 3x = 6x and 2 × (-7) = -14, so we get 6x - 14 ≤ 5x + 4 Subtract 5x from both sides: 6x - 5x - 14 ≤ 5x - 5x + 4, which gives x - 14 ≤ 4 Add 14 to both sides: x - 14 + 14 ≤ 4 + 14, which gives x ≤ 18 The solution is x ≤ 18
   Full step-by-step solution

   Step 1: Distribute the 2: 2 × 3x = 6x and 2 × (-7) = -14, so we get 6x - 14 ≤ 5x + 4 Step 2: Subtract 5x from both sides: 6x - 5x - 14 ≤ 5x - 5x + 4, which gives x - 14 ≤ 4 Step 3: Add 14 to both sides: x - 14 + 14 ≤ 4 + 14, which gives x ≤ 18 Step 4: The solution is x ≤ 18

8. Tom is saving money to buy a video game that costs $53. Tom already has $19 saved and plans to save $6 each week from an allowance. Let x represent the number of weeks Tom needs to save. Write and solve an inequality to find how many weeks are needed. Then describe how to graph the solution on a number line. Answer: 6 Solution: Let x = number of weeks. Total saved = 19 + 6x.
   Full step-by-step solution

   Step 1: Let x = number of weeks. Step 2: Total saved = 19 + 6x. Need at least 53: 19 + 6x ≥ 53 Step 3: Subtract 19: 6x ≥ 53 - 19 Step 4: Divide by 6: x ≥ (53 - 19) / 6 Step 5: Since x must be a whole number of weeks, x ≥ 6 Step 6: On a number line, place a closed circle at 6 and shade to the right, showing that 6 or more weeks are needed.