Compound Inequalities

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Answer Key & Explanations

Compound Inequalities · Grade 7 · Worksheet 3

1. Matiu is helping his school's outdoor education program design a new obstacle course. The budget allows for a rope course that must have a total length between 150 and 300 meters. The course will have 3 sections of equal length. Additionally, each section must be at least 30 meters long but no more than 80 meters long to meet safety requirements. Let x represent the length of each section in meters. Write and solve a compound inequality to find all possible lengths for each section that satisfy both the budget and safety requirements. Answer: 50 < x < 80 Solution: The total length of the course is 3x. The budget requires 150 < 3x < 300. Step 2: Solve the budget inequality: Divide all parts by 3: 150/3 < 3x/3 < 300/3, so 50 < x < 100.
   Full step-by-step solution

   Step 1: The total length of the course is 3x. The budget requires 150 < 3x < 300. Step 2: Solve the budget inequality: Divide all parts by 3: 150/3 < 3x/3 < 300/3, so 50 < x < 100. Step 3: The safety requirement says each section must be at least 30 m and no more than 80 m: 30 < x < 80. Step 4: Both conditions must be true (AND), so we take the overlap of 50 < x < 100 and 30 < x < 80. Step 5: The more restrictive lower bound is 50 (since 50 > 30), and the more restrictive upper bound is 80 (since 80 < 100). Step 6: The compound inequality is 50 < x < 80. So each section must be longer than 50 m but shorter than 80 m to satisfy both the budget and safety requirements.

2. 3x + 7 > 16 and 2x - 5 ≤ 9 Answer: 3 < x ≤ 7 Solution: 1) 3x + 7 > 16 2) 2x - 5 ≤ 9 Solve the first inequality 3x + 7 > 16 Subtract 7 from both sides: 3x + 7 - 7 > 16 - 7 3x > 9 Divide both sides by 3: x > 3 So from the first inequality, x must be greater than 3.
   Full step-by-step solution

   Let's solve the compound inequality step by step. We have two inequalities: 1) 3x + 7 > 16 2) 2x - 5 ≤ 9 --- **Step 1: Solve the first inequality 3x + 7 > 16** Subtract 7 from both sides: 3x + 7 - 7 > 16 - 7 3x > 9 Divide both sides by 3: x > 3 So from the first inequality, x must be greater than 3. --- **Step 2: Solve the second inequality 2x - 5 ≤ 9** Add 5 to both sides: 2x - 5 + 5 ≤ 9 + 5 2x ≤ 14 Divide both sides by 2: x ≤ 7 So from the second inequality, x must be less than or equal to 7. --- **Step 3: Combine the results** From Step 1: x > 3 From Step 2: x ≤ 7 We need x to satisfy both conditions at the same time. So x must be greater than 3 AND less than or equal to 7. That is written as: 3 < x ≤ 7 --- **Final Answer:** 3 < x ≤ 7

3. Liam is organizing a school field day event and needs to set up a volleyball net. The net's height must be at least 220 centimeters but no more than 250 centimeters above the ground to meet official regulations. The net is attached to two poles, and each pole has an adjustable section that can extend the pole's height by x centimeters. The fixed part of each pole is 185 centimeters tall. Write and solve a compound inequality to find all possible extension lengths x (in centimeters) that will keep the net height within the required range. Answer: 35 ≤ x ≤ 65 Solution: The total height of the pole is the fixed height (185 cm) plus the extension (x cm). So total height = 185 + x. The net height must be at least 220 cm and no more than 250 cm.
   Full step-by-step solution

   Step 1: The total height of the pole is the fixed height (185 cm) plus the extension (x cm). So total height = 185 + x. Step 2: The net height must be at least 220 cm and no more than 250 cm. This gives the compound inequality: 220 ≤ 185 + x ≤ 250. Step 3: Subtract 185 from all three parts: 220 - 185 ≤ 185 + x - 185 ≤ 250 - 185. Step 4: Simplify: 35 ≤ x ≤ 65. Step 5: This means the extension length must be between 35 cm and 65 cm inclusive. Final answer: 35 ≤ x ≤ 65.

4. Maya is designing a science fair display board that must have an area between 1,200 and 1,800 square inches. The board's length must be at least 40 inches but no more than 60 inches. If the width is 30 inches, write a compound inequality to represent all possible lengths that satisfy both the area and length constraints. Answer: 40 ≤ L ≤ 60 Solution: When working with compound inequalities for area constraints, you need to consider both the direct length restrictions and the area formula.
   Full step-by-step solution

   When working with compound inequalities for area constraints, you need to consider both the direct length restrictions and the area formula. The area is calculated by multiplying length and width, so with a fixed width, the length directly determines the area. You need to find values that satisfy all given conditions simultaneously.

5. 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.

6. Charlotte is helping her school's robotics team build a drone that can safely carry a payload. The drone's total takeoff weight, including the payload, must be at least 1,247 grams to maintain stability in wind, but no more than 1,832 grams to avoid exceeding motor capacity. The drone itself weighs 1,020 grams. If p represents the weight of the payload in grams, write a compound inequality for p, then solve it to find the range of possible payload weights that will keep the drone safely in the air. Answer: 227 ≤ p ≤ 812 Solution: Write the total weight as drone weight plus payload: 1020 + p The total must be at least 1247 grams: 1020 + p ≥ 1247 The total must be no more than 1832 grams: 1020 + p ≤ 1832 Solve the first inequality: 1020 + p ≥ 1247 → p ≥ 1247 - 1020 → p ≥ 227 Solve the second inequality: 1020 + p ≤ 1832 → p…
   Full step-by-step solution

   Step 1: Write the total weight as drone weight plus payload: 1020 + p Step 2: The total must be at least 1247 grams: 1020 + p ≥ 1247 Step 3: The total must be no more than 1832 grams: 1020 + p ≤ 1832 Step 4: Solve the first inequality: 1020 + p ≥ 1247 → p ≥ 1247 - 1020 → p ≥ 227 Step 5: Solve the second inequality: 1020 + p ≤ 1832 → p ≤ 1832 - 1020 → p ≤ 812 Step 6: Combine both inequalities: 227 ≤ p ≤ 812 Step 7: This means the payload must weigh at least 227 grams and at most 812 grams. The answer is 227 ≤ p ≤ 812.