Graph Inequalities

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Answer Key & Explanations

Graph Inequalities · Grade 7 · Worksheet 3

1. Emma is graphing the solution to the inequality 3x - 7 ≤ 11 on a number line. Solve the inequality, then describe the graph. What number is at the boundary point of the solution? (Give the boundary number.) Answer: 6 Solution: Start with the inequality: 3x - 7 ≤ 11. Add 7 to both sides to isolate the term with x: 3x - 7 + 7 ≤ 11 + 7 → 3x ≤ 18. Divide both sides by 3 to solve for x: (3x)/3 ≤ 18/3 → x ≤ 6.
   Full step-by-step solution

   Step 1: Start with the inequality: 3x - 7 ≤ 11. Step 2: Add 7 to both sides to isolate the term with x: 3x - 7 + 7 ≤ 11 + 7 → 3x ≤ 18. Step 3: Divide both sides by 3 to solve for x: (3x)/3 ≤ 18/3 → x ≤ 6. Step 4: The boundary point is the number where x equals the boundary, which is 6. On the number line, a closed circle is placed at 6 and the line is shaded to the left, representing all numbers less than or equal to 6. Final answer: 6.

2. Noah is graphing the solution to the inequality 4x - 9 > 19 on a number line. He needs to draw an open or closed circle and shade in the correct direction. Describe the number line graph Noah should draw, including the type of circle and the direction of shading. Answer: Open circle at 7, shading to the right Solution: Solve the inequality 4x - 9 > 19. Add 9 to both sides: 4x - 9 + 9 > 19 + 9, so 4x > 28. Divide both sides by 4: 4x / 4 > 28 / 4, so x > 7.
   Full step-by-step solution

   Step 1: Solve the inequality 4x - 9 > 19. Add 9 to both sides: 4x - 9 + 9 > 19 + 9, so 4x > 28. Divide both sides by 4: 4x / 4 > 28 / 4, so x > 7. Step 2: Interpret the inequality x > 7. The solution includes all numbers greater than 7, but not 7 itself. Step 3: Draw the number line graph. - Place a circle at 7. Since 7 is NOT included (x > 7, not x >= 7), use an open circle. - Shade the number line to the right of 7, because x is greater than 7. Final answer: Open circle at 7, shading to the right.

3. Graph 5(2x - 15) + 20 ≥ 10x - 55 Answer: All real numbers (shade entire number line) Solution: Distribute the 5 on the left side: 5(2x - 15) = 10x - 75. The inequality becomes: 10x - 75 + 20 ≥ 10x - 55. Combine like terms on the left: 10x - 55 ≥ 10x - 55.
   Full step-by-step solution

   Step 1: Distribute the 5 on the left side: 5(2x - 15) = 10x - 75. The inequality becomes: 10x - 75 + 20 ≥ 10x - 55. Step 2: Combine like terms on the left: 10x - 55 ≥ 10x - 55. Step 3: Subtract 10x from both sides: 10x - 55 - 10x ≥ 10x - 55 - 10x → -55 ≥ -55. Step 4: The statement -55 ≥ -55 is always true. This means the inequality holds for every value of x. Step 5: The solution is all real numbers. On a number line, shade the entire line from left to right (no open or closed circle needed).

4. Matiu is checking the temperature in his freezer. The temperature, t, in degrees Celsius, must satisfy the inequality 3t - 18 ≤ -45. Graph the solution to this inequality on a number line. Describe the graph by stating the endpoint value, whether the circle is open or closed, and the direction of the shading. Answer: Closed circle at -9, shading to the left Solution: Start with the inequality: 3t - 18 ≤ -45 Add 18 to both sides: 3t - 18 + 18 ≤ -45 + 18 → 3t ≤ -27 Divide both sides by 3 (positive, so inequality sign stays the same): t ≤ -9 The solution is all numbers less than or equal to -9.
   Full step-by-step solution

   Step 1: Start with the inequality: 3t - 18 ≤ -45 Step 2: Add 18 to both sides: 3t - 18 + 18 ≤ -45 + 18 → 3t ≤ -27 Step 3: Divide both sides by 3 (positive, so inequality sign stays the same): t ≤ -9 Step 4: The solution is all numbers less than or equal to -9. On a number line: place a closed circle at -9 (because ≤ includes -9) and shade to the left (toward smaller numbers). The answer is: closed circle at -9, shading to the left.

5. Emma is organizing a school fundraiser and needs to buy supplies. She has a budget of $500. She needs to buy at least 100 t-shirts that cost $4 each and some banners that cost $25 each. Write an inequality to represent the maximum number of banners she can buy, then solve for the actual maximum number of banners. Answer: 4 Solution: Calculate the cost of the required t-shirts: 100 t-shirts × $4 per t-shirt = $400 Subtract the t-shirt cost from the total budget: $500 - $400 = $100 remaining Divide the remaining money by the cost per banner: $100 ÷ $25 per banner = 4 banners Write the inequality: Let b represent the number of…
   Full step-by-step solution

   Step 1: Calculate the cost of the required t-shirts: 100 t-shirts × $4 per t-shirt = $400 Step 2: Subtract the t-shirt cost from the total budget: $500 - $400 = $100 remaining Step 3: Divide the remaining money by the cost per banner: $100 ÷ $25 per banner = 4 banners Step 4: Write the inequality: Let b represent the number of banners. The inequality is 4(100) + 25b ≤ 500 Step 5: Solve the inequality: 400 + 25b ≤ 500 → 25b ≤ 100 → b ≤ 4 The maximum number of banners Emma can buy is 4.

6. Emma is organizing a school fundraiser and needs to buy supplies. She has a budget of $500. She needs to buy at least 50 t-shirts that cost $8 each and some banners that cost $15 each. Write an inequality to represent all possible numbers of banners Emma can buy without exceeding her budget, then determine the maximum number of banners she can purchase. Answer: 6 Solution: Calculate the cost of the required t-shirts: 50 t-shirts × $8 per t-shirt = $400 Subtract this from the total budget to find remaining money: $500 - $400 = $100 Let b represent the number of banners.
   Full step-by-step solution

   Step 1: Calculate the cost of the required t-shirts: 50 t-shirts × $8 per t-shirt = $400 Step 2: Subtract this from the total budget to find remaining money: $500 - $400 = $100 Step 3: Let b represent the number of banners. Each banner costs $15, so the inequality is: 15b ≤ 100 Step 4: Solve for b: b ≤ 100 ÷ 15 Step 5: 100 ÷ 15 = 6.666... Step 6: Since Emma can't buy a fraction of a banner, the maximum whole number is 6. The maximum number of banners Emma can buy is 6.

7. Liam is designing a rectangular garden with a length of 12 meters. The perimeter of the garden must be at least 40 meters but no more than 52 meters. Write a compound inequality to represent all possible widths, w, that Liam can use for his garden. Answer: 8 ≤ w ≤ 14 Solution: Perimeter = 2 × length + 2 × width Given length = 12 m, width = w m. Perimeter = 2(12) + 2(w) = 24 + 2w. The perimeter must be at least 40 m but no more than 52 m.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Recall the perimeter formula for a rectangle** Perimeter = 2 × length + 2 × width Given length = 12 m, width = w m. Perimeter = 2(12) + 2(w) = 24 + 2w. --- **Step 2: Set up the compound inequality from the problem** The perimeter must be at least 40 m but no more than 52 m. So: 40 ≤ Perimeter ≤ 52 Substitute Perimeter = 24 + 2w: 40 ≤ 24 + 2w ≤ 52 --- **Step 3: Isolate w in the compound inequality** We can solve all parts at once: Subtract 24 from all three parts: 40 − 24 ≤ 24 + 2w − 24 ≤ 52 − 24 16 ≤ 2w ≤ 28 --- **Step 4: Divide all parts by 2** 16/2 ≤ 2w/2 ≤ 28/2 8 ≤ w ≤ 14 --- **Step 5: Interpret the result** The possible widths w (in meters) that satisfy the perimeter condition are between 8 and 14 inclusive. --- **Final Answer:** 8 ≤ w ≤ 14

8. Graph 5(2x - 9) + 3 ≥ 8x - 12 Answer: x ≥ 15 Solution: Distribute the 5 on the left side: 5(2x - 9) = 10x - 45. The inequality becomes: 10x - 45 + 3 ≥ 8x - 12. Combine like terms on the left: -45 + 3 = -42, so we have 10x - 42 ≥ 8x - 12.
   Full step-by-step solution

   Step 1: Distribute the 5 on the left side: 5(2x - 9) = 10x - 45. The inequality becomes: 10x - 45 + 3 ≥ 8x - 12. Step 2: Combine like terms on the left: -45 + 3 = -42, so we have 10x - 42 ≥ 8x - 12. Step 3: Subtract 8x from both sides: 10x - 8x - 42 ≥ 8x - 8x - 12 → 2x - 42 ≥ -12. Step 4: Add 42 to both sides: 2x - 42 + 42 ≥ -12 + 42 → 2x ≥ 30. Step 5: Divide both sides by 2: 2x/2 ≥ 30/2 → x ≥ 15. The solution is x ≥ 15. On a number line, draw a closed circle at 15 and shade to the right.