Compound Inequalities

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Compound Inequalities · Grade 7 · Worksheet 1

1. A science lab is testing a new chemical compound that must be stored at a temperature between -12°C and 8°C to remain stable. The lab's freezer can maintain temperatures from -20°C to 5°C. Write a compound inequality that represents the temperature range (t) where the chemical can be safely stored in this freezer. Answer: -12 ≤ t ≤ 5 Solution: The chemical needs temperatures between -12°C and 8°C, so -12 ≤ t ≤ 8 The freezer can maintain temperatures between -20°C and 5°C, so -20 ≤ t ≤ 5 To find where both conditions are true, we need the overlapping range The lower limit is the higher of the two lower bounds: max(-12, -20) = -12 The…
   Full step-by-step solution

   Step 1: The chemical needs temperatures between -12°C and 8°C, so -12 ≤ t ≤ 8 Step 2: The freezer can maintain temperatures between -20°C and 5°C, so -20 ≤ t ≤ 5 Step 3: To find where both conditions are true, we need the overlapping range Step 4: The lower limit is the higher of the two lower bounds: max(-12, -20) = -12 Step 5: The upper limit is the lower of the two upper bounds: min(8, 5) = 5 Step 6: Therefore, the safe temperature range is -12 ≤ t ≤ 5 The answer is -12 ≤ t ≤ 5.

2. A science lab is testing a new chemical that must be kept at a temperature between -12°C and 48°C to remain stable. The lab's cooling system can adjust the temperature by exactly 7°C from the current room temperature of 22°C. Write a compound inequality to represent all the safe temperature adjustments, t, the cooling system can make. Answer: -34 ≤ t ≤ 26 Solution: When working with temperature adjustments, you need to consider how adding or subtracting from the starting temperature affects the final value.
   Full step-by-step solution

   When working with temperature adjustments, you need to consider how adding or subtracting from the starting temperature affects the final value. The safe range represents boundaries that the final temperature cannot exceed, so you need to determine what adjustments would keep the result within those boundaries.

3. Mason is drawing a number line graph for a compound inequality. He shades all points to the right of 9 and all points to the left of 21, with solid circles at both 9 and 21. The shaded region represents all numbers that satisfy the inequality. Write the compound inequality that matches Mason's graph using x as the variable. Answer: x ≥ 9 AND x ≤ 21 Solution: Solid circles at 9 and 21 mean the endpoints are included, so we use ≥ and ≤. Shading to the right of 9 means x is greater than or equal to 9: x ≥ 9. Shading to the left of 21 means x is less than or equal to 21: x ≤ 21.
   Full step-by-step solution

   Step 1: Solid circles at 9 and 21 mean the endpoints are included, so we use ≥ and ≤. Step 2: Shading to the right of 9 means x is greater than or equal to 9: x ≥ 9. Step 3: Shading to the left of 21 means x is less than or equal to 21: x ≤ 21. Step 4: Since both conditions must be true at the same time, we use AND to combine them. Step 5: The compound inequality is x ≥ 9 AND x ≤ 21. The answer is x ≥ 9 AND x ≤ 21.

4. 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.

5. Charlotte is helping her school's environmental club plant a rectangular garden. The garden's length must be at least 12 feet but less than 27 feet, and its width must be more than 7 feet but no more than 17 feet. The area of the garden is calculated by multiplying length and width. Write a compound inequality that represents all possible areas, A, in square feet, that the garden could have. Answer: 84 < A < 459 Solution: Identify the range for length (L): 12 ≤ L < 27 Identify the range for width (W): 7 < W ≤ 17 Find the minimum possible area: Use the smallest length (12) and smallest width (7). Minimum area = 12 × 7 = 84.
   Full step-by-step solution

   Step 1: Identify the range for length (L): 12 ≤ L < 27 Step 2: Identify the range for width (W): 7 < W ≤ 17 Step 3: Find the minimum possible area: Use the smallest length (12) and smallest width (7). Minimum area = 12 × 7 = 84. Since the length can be exactly 12 but the width must be more than 7, the area must be greater than 84, so A > 84. Step 4: Find the maximum possible area: Use the largest length (27) and largest width (17). Maximum area = 27 × 17 = 459. Since the length must be less than 27 and the width can be exactly 17, the area must be less than 459, so A < 459. Step 5: Combine the inequalities: 84 < A < 459. The compound inequality is 84 < A < 459.

6. 3(2x - 5) + 7 ≥ 4(x + 3) - 1 Answer: x ≥ 5.5 Solution: When solving inequalities with parentheses, distribute first to eliminate them. Then combine constant terms and variable terms separately on each side.
   Full step-by-step solution

   When solving inequalities with parentheses, distribute first to eliminate them. Then combine constant terms and variable terms separately on each side. Remember that when multiplying or dividing by a negative number, you must flip the inequality sign, but this doesn't apply to addition or subtraction.

7. Noah is designing a rectangular skatepark that must have a length between 25 and 40 meters. The width must be at least 15 meters but less than 22 meters. If the area of the skatepark is calculated by multiplying length and width, write a compound inequality that represents all possible areas, A, in square meters, that the skatepark could have. Answer: 375 < A < 880 Solution: Identify the range for length (L): 25 < L < 40 Identify the range for width (W): 15 ≤ W < 22 Find the minimum possible area: Use the smallest length (25) and smallest width (15) Minimum area = 25 × 15 = 375 Since both length and width can be greater than their minimums, the area must be greater…
   Full step-by-step solution

   Step 1: Identify the range for length (L): 25 < L < 40 Step 2: Identify the range for width (W): 15 ≤ W < 22 Step 3: Find the minimum possible area: Use the smallest length (25) and smallest width (15) Minimum area = 25 × 15 = 375 Since both length and width can be greater than their minimums, the area must be greater than 375, so A > 375 Step 4: Find the maximum possible area: Use the largest length (40) and largest width (22) Maximum area = 40 × 22 = 880 Since the width must be less than 22 and length less than 40, the area must be less than 880, so A < 880 Step 5: Combine the inequalities: 375 < A < 880 The compound inequality is 375 < A < 880.