Compound Inequalities

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Compound Inequalities · Grade 7 · Worksheet 2

1. Emma is designing a science fair display that needs to be between 120 and 180 square inches in area. The display must be rectangular with a width of exactly 15 inches. Write a compound inequality to represent all possible lengths, in inches, that the display can have. Answer: 8 < L < 12 Solution: We know width = 15 inches and area must be between 120 and 180 square inches Write the inequality: 120 < 15L < 180 Divide all parts of the inequality by 15: 120 ÷ 15 < L < 180 ÷ 15 Calculate: 8 < L < 12 The length must be greater than 8 inches and less than 12 inches Final answer: 8 < L < 12
   Full step-by-step solution

   Step 1: The area of a rectangle is length × width Step 2: We know width = 15 inches and area must be between 120 and 180 square inches Step 3: Write the inequality: 120 < 15L < 180 Step 4: Divide all parts of the inequality by 15: 120 ÷ 15 < L < 180 ÷ 15 Step 5: Calculate: 8 < L < 12 Step 6: The length must be greater than 8 inches and less than 12 inches Final answer: 8 < L < 12

2. 2(x + 3) - 5 < 7 and 3x - 4 > 2 = ? Answer: 2 < x < 3 Solution: 1) 2(x + 3) - 5 < 7 2) 3x - 4 > 2 2(x + 3) - 5 < 7 First, distribute the 2: 2x + 6 - 5 < 7 2x + 1 < 7 Subtract 1 from both sides: 2x < 6 Divide both sides by 2: x < 3 So from the first inequality: x < 3.
   Full step-by-step solution

   Let's solve the compound inequality step by step. We have two inequalities: 1) 2(x + 3) - 5 < 7 2) 3x - 4 > 2 --- **Step 1: Solve the first inequality** 2(x + 3) - 5 < 7 First, distribute the 2: 2x + 6 - 5 < 7 Simplify: 2x + 1 < 7 Subtract 1 from both sides: 2x < 6 Divide both sides by 2: x < 3 So from the first inequality: x < 3. --- **Step 2: Solve the second inequality** 3x - 4 > 2 Add 4 to both sides: 3x > 6 Divide both sides by 3: x > 2 So from the second inequality: x > 2. --- **Step 3: Combine the results** From the first inequality: x < 3 From the second inequality: x > 2 Combining: 2 < x < 3 --- **Final answer:** 2 < x < 3

3. Aisha is designing a community garden with a rectangular plot that must be at least 15 feet wide but no more than 25 feet wide. The length of the plot must be at least 30 feet longer than the width, but the total area cannot exceed 1,500 square feet. If the width is represented by w and the length by l, which of the following systems of inequalities correctly models the constraints for Aisha's garden plot? Answer: w ≥ 15, w ≤ 25, l ≥ w + 30, w * l ≤ 1500 Solution: - Width = w - Length = l - "at least 15 feet wide" means w ≥ 15 - "no more than 25 feet wide" means w ≤ 25 w ≥ 15 w ≤ 25 "Length must be at least 30 feet longer than the width" means: l ≥ w + 30 "Total area cannot exceed 1,500 square feet" means: Area = w × l ≤ 1500 w × l ≤ 1500 1.
   Full step-by-step solution

   Let's break down the problem step by step. --- **Step 1: Understand the variables** We have: - Width = w - Length = l --- **Step 2: Width constraints** The problem says: - "at least 15 feet wide" means w ≥ 15 - "no more than 25 feet wide" means w ≤ 25 So: w ≥ 15 w ≤ 25 --- **Step 3: Length constraint relative to width** "Length must be at least 30 feet longer than the width" means: l ≥ w + 30 --- **Step 4: Area constraint** "Total area cannot exceed 1,500 square feet" means: Area = w × l ≤ 1500 So: w × l ≤ 1500 --- **Step 5: Combine all constraints** We have: 1. w ≥ 15 2. w ≤ 25 3. l ≥ w + 30 4. w × l ≤ 1500 That matches the given correct answer. --- **Step 6: Check reasoning with example numbers** Let’s test with w = 15: - l ≥ 15 + 30 = 45 - Area = 15 × 45 = 675 ≤ 1500 ✅ Let’s test with w = 25: - l ≥ 25 + 30 = 55 - Area = 25 × 55 = 1375 ≤ 1500 ✅ What if l is larger? Say w = 25, l = 60: Area = 25 × 60 = 1500 ✅ still okay. But if w = 25, l = 61: Area = 1525 ❌ violates w × l ≤ 1500. So the system correctly models all constraints. --- **Final answer:** w ≥ 15, w ≤ 25, l ≥ w + 30, w * l ≤ 1500

4. A number line is drawn from -15 to 25. A shaded region is shown between two points. The inequality represented by the graph is: "x is greater than or equal to -7 AND x is less than 13." Write the solution as a compound inequality in the form a ≤ x < b, then graph it on a number line. What are the values of a and b? Answer: a = -7, b = 13 Solution: Translate the words into symbols. 'x is greater than or equal to -7' means x ≥ -7. 'x is less than 13' means x < 13.
   Full step-by-step solution

   Step 1: Translate the words into symbols. 'x is greater than or equal to -7' means x ≥ -7. 'x is less than 13' means x < 13. Step 2: Combine them using AND. Since both must be true at the same time, we write -7 ≤ x < 13. Step 3: On a number line from -15 to 25, place a closed circle at -7 and an open circle at 13. Shade the number line between them. Step 4: The values are a = -7 and b = 13. The answer is a = -7, b = 13.

5. Emma is designing a rectangular garden on a coordinate grid, where each unit represents 1 foot. The garden's corners are at (0, 0), (30, 0), (30, 20), and (0, 20). She wants to plant flowers in a triangular flower bed with vertices at (0, 0), (15, 0), and (0, 10). The rest of the garden will be grass. What is the area, in square feet, of the grass portion of the garden? Answer: 525 Solution: Find the area of the rectangular garden. Length = 30 feet, Width = 20 feet. Area = 30 * 20 = 600 square feet.
   Full step-by-step solution

   Step 1: Find the area of the rectangular garden. Length = 30 feet, Width = 20 feet. Area = 30 * 20 = 600 square feet. Step 2: Find the area of the triangular flower bed. The triangle has vertices at (0, 0), (15, 0), and (0, 10). This is a right triangle with base = 15 feet and height = 10 feet. Area of triangle = 1/2 * base * height = 1/2 * 15 * 10 = 1/2 * 150 = 75 square feet. Step 3: Subtract the flower bed area from the total garden area to find the grass area. Grass area = 600 - 75 = 525 square feet. The answer is 525.

6. 4(2x - 6) + 8 ≥ 40 and 6x - 14 < 58 = ? Answer: 7 ≤ x < 12 Solution: Solve the first inequality: 4(2x - 6) + 8 ≥ 40 4(2x - 6) + 8 ≥ 40 8x - 24 + 8 ≥ 40 8x - 16 ≥ 40 8x ≥ 56 x ≥ 7 Solve the second inequality: 6x - 14 < 58 6x - 14 < 58 6x < 72 x < 12 Combine the solutions: x ≥ 7 AND x < 12 This means x is greater than or equal to 7, but less than 12.
   Full step-by-step solution

   Step 1: Solve the first inequality: 4(2x - 6) + 8 ≥ 40 4(2x - 6) + 8 ≥ 40 8x - 24 + 8 ≥ 40 8x - 16 ≥ 40 8x ≥ 56 x ≥ 7 Step 2: Solve the second inequality: 6x - 14 < 58 6x - 14 < 58 6x < 72 x < 12 Step 3: Combine the solutions: x ≥ 7 AND x < 12 This means x is greater than or equal to 7, but less than 12. Step 4: Write the compound inequality: 7 ≤ x < 12 The final answer is 7 ≤ x < 12.

7. Mason is saving up to buy a new bicycle that costs between $250 and $375. He already has $85 saved. He plans to mow lawns over the summer and will earn $15 per lawn. Write and solve a compound inequality to find the number of lawns, n, Mason must mow to have enough money to buy the bicycle. Answer: 11 < n < 20 Solution: Mason already has $85. He earns $15 per lawn. After mowing n lawns, his total money is 85 + 15n.
   Full step-by-step solution

   Step 1: Mason already has $85. He earns $15 per lawn. After mowing n lawns, his total money is 85 + 15n. Step 2: The bike costs between $250 and $375. This means the total money must be at least $250 and at most $375. Step 3: Write the compound inequality: 250 ≤ 85 + 15n ≤ 375. Step 4: Subtract 85 from all three parts: 250 - 85 ≤ 15n ≤ 375 - 85, which gives 165 ≤ 15n ≤ 290. Step 5: Divide all three parts by 15: 165/15 ≤ n ≤ 290/15, which simplifies to 11 ≤ n ≤ 19.333... Step 6: Since n must be a whole number (he can't mow a fraction of a lawn), we round: n must be at least 11 lawns and at most 19 lawns. But because the upper bound is strict (bike costs less than $375, not equal to $375), we use inequality signs carefully. For the upper bound, if n = 19, total = 85 + 15(19) = 85 + 285 = 370, which is less than 375. If n = 20, total = 85 + 300 = 385, which is over $375. So n must be less than 20. For the lower bound, if n = 10, total = 85 + 150 = 235, which is less than $250. If n = 11, total = 85 + 165 = 250, which works. So n must be greater than 10. Thus the compound inequality in whole numbers is: 11 ≤ n < 20. The answer is 11 < n < 20 (when considering n as a real number).