Compound Inequalities

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Answer Key & Explanations

Compound Inequalities · Grade 7 · Worksheet 3

1. 2x + 5 > 11 and 3x - 4 ≤ 17 Answer: 3 < x ≤ 7 Solution: 1) 2x + 5 > 11 2) 3x - 4 ≤ 17 Solve the first inequality 2x + 5 > 11 Subtract 5 from both sides: 2x + 5 - 5 > 11 - 5 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) 2x + 5 > 11 2) 3x - 4 ≤ 17 --- **Step 1: Solve the first inequality 2x + 5 > 11** Subtract 5 from both sides: 2x + 5 - 5 > 11 - 5 2x > 6 Divide both sides by 2: x > 3 So from the first inequality: x > 3. --- **Step 2: Solve the second inequality 3x - 4 ≤ 17** Add 4 to both sides: 3x - 4 + 4 ≤ 17 + 4 3x ≤ 21 Divide both sides by 3: x ≤ 7 So from the second inequality: x ≤ 7. --- **Step 3: Combine the two inequalities** From Step 1: x > 3 From Step 2: x ≤ 7 Putting them together: 3 < x ≤ 7 --- **Step 4: Interpret the result** This means x is greater than 3 and less than or equal to 7. In interval notation, this would be (3, 7] but we write it as 3 < x ≤ 7. --- **Final Answer:** 3 < x ≤ 7

2. Hana is designing a rectangular garden on a coordinate grid. The garden has vertices at (0, 0), (24, 0), (24, 16), and (0, 16). She wants to build a square flower bed inside the garden with vertices at (4, 4), (12, 4), (12, 12), and (4, 12). What is the area of the garden that is NOT covered by the flower bed? Answer: 320 Solution: Find the area of the rectangular garden. The length is 24 units and the width is 16 units. Area = length × width = 24 × 16 = 384 square units.
   Full step-by-step solution

   Step 1: Find the area of the rectangular garden. The length is 24 units and the width is 16 units. Area = length × width = 24 × 16 = 384 square units. Step 2: Find the area of the square flower bed. The side length is the distance from (4,4) to (12,4), which is 8 units. Area = side × side = 8 × 8 = 64 square units. Step 3: Subtract the flower bed area from the garden area: 384 - 64 = 320 square units. The answer is 320.

3. Liam is designing a community garden with a rectangular plot. The length of the plot must be at least 15 meters but less than 25 meters. The width must be more than 8 meters but no more than 12 meters. If the area of the garden is calculated by multiplying the length and width, what is the range of possible areas for the garden? Answer: The area is greater than 120 square meters and less than or equal to 300 square meters. Solution: When working with compound inequalities for area, you need to consider the combinations that give you the smallest and largest possible products.
   Full step-by-step solution

   When working with compound inequalities for area, you need to consider the combinations that give you the smallest and largest possible products. The minimum area comes from the smallest length and smallest width, while the maximum area comes from the largest length and largest width. However, you must be careful with the inequality signs to determine whether the endpoints are included or excluded from the final range.

4. Matiu is helping his family plan a road trip. The car's fuel tank can hold between 45 and 60 liters of gas. Currently, the tank has 12 liters of gas. Matiu wants to add some additional gas, g liters, so that the total amount of gas in the tank stays within the safe capacity range. Write a compound inequality that represents all possible amounts of gas, g, that Matiu can add to the tank. Answer: 33 < g < 48 Solution: The tank can hold between 45 and 60 liters, so the total gas after adding g liters must satisfy: 45 < total < 60. The current gas is 12 liters, so total = 12 + g. Write the compound inequality: 45 < 12 + g < 60.
   Full step-by-step solution

   Step 1: The tank can hold between 45 and 60 liters, so the total gas after adding g liters must satisfy: 45 < total < 60. Step 2: The current gas is 12 liters, so total = 12 + g. Step 3: Write the compound inequality: 45 < 12 + g < 60. Step 4: Solve the left part: 45 < 12 + g → subtract 12 from both sides: 45 - 12 < g → 33 < g. Step 5: Solve the right part: 12 + g < 60 → subtract 12 from both sides: g < 60 - 12 → g < 48. Step 6: Combine: 33 < g < 48. The compound inequality is 33 < g < 48.

5. 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.

6. 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

7. 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