Compound Inequalities

lessonbunny.com

Answer Key & Explanations

Compound Inequalities · Grade 7 · Worksheet 2

1. Emma is saving money to buy a new bicycle that costs between $195 and $265. She already has $75 saved. She earns $15 each week from walking neighbors' dogs. Write a compound inequality to find the range of whole weeks, w, she needs to save in order to afford the bicycle. Answer: 8 ≤ w ≤ 12 Solution: Emma's total savings after w weeks = 75 + 15w This total must be at least $195 and at most $265.
   Full step-by-step solution

   Step 1: Emma's total savings after w weeks = 75 + 15w Step 2: This total must be at least $195 and at most $265. Step 3: Write the compound inequality: 195 ≤ 75 + 15w ≤ 265 Step 4: Subtract 75 from all three parts: 195 - 75 ≤ 75 + 15w - 75 ≤ 265 - 75 Step 5: Simplify: 120 ≤ 15w ≤ 190 Step 6: Divide all three parts by 15: 120/15 ≤ w ≤ 190/15 Step 7: Simplify: 8 ≤ w ≤ 12.67 Step 8: Since w must be a whole number of weeks, w must be at least 8 and at most 12. Final answer: 8 ≤ w ≤ 12

2. 2(3x - 5) + 7 ≥ 15 and 4x - 3 < 17 = ? Answer: 4 ≤ x < 5 Solution: Solve the first inequality: 2(3x - 5) + 7 ≥ 15 2(3x - 5) + 7 ≥ 15 6x - 10 + 7 ≥ 15 6x - 3 ≥ 15 6x ≥ 18 x ≥ 3 Solve the second inequality: 4x - 3 < 17 4x - 3 < 17 4x < 20 x < 5 Combine the solutions: x ≥ 3 and x < 5 This means 3 ≤ x < 5 Check if we can simplify: 3 ≤ x < 5 is the same as 4 ≤ x < 5…
   Full step-by-step solution

   Step 1: Solve the first inequality: 2(3x - 5) + 7 ≥ 15 2(3x - 5) + 7 ≥ 15 6x - 10 + 7 ≥ 15 6x - 3 ≥ 15 6x ≥ 18 x ≥ 3 Step 2: Solve the second inequality: 4x - 3 < 17 4x - 3 < 17 4x < 20 x < 5 Step 3: Combine the solutions: x ≥ 3 and x < 5 This means 3 ≤ x < 5 Step 4: Check if we can simplify: 3 ≤ x < 5 is the same as 4 ≤ x < 5 when considering integer solutions, but for all real numbers, the solution is 3 ≤ x < 5. The final answer is 3 ≤ x < 5.

3. 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.

4. 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.

5. 2(x + 3) > 8 and 3x - 5 ≤ 10 Answer: 1 < x ≤ 5 Solution: 2(x + 3) > 8 and 3x - 5 ≤ 10 2(x + 3) > 8 Divide both sides by 2: x + 3 > 4 Subtract 3 from both sides: x > 1 So the first inequality means: x > 1.
   Full step-by-step solution

   Let's solve the system of inequalities step by step. We have: 2(x + 3) > 8 and 3x - 5 ≤ 10 --- **Step 1: Solve the first inequality** 2(x + 3) > 8 Divide both sides by 2: x + 3 > 4 Subtract 3 from both sides: x > 1 So the first inequality means: x > 1. --- **Step 2: Solve the second inequality** 3x - 5 ≤ 10 Add 5 to both sides: 3x ≤ 15 Divide both sides by 3: x ≤ 5 So the second inequality means: x ≤ 5. --- **Step 3: Combine the solutions** From the first inequality: x > 1 From the second inequality: x ≤ 5 We need x to satisfy both conditions at the same time. So x must be greater than 1 AND less than or equal to 5. That is: 1 < x ≤ 5 --- **Final Answer:** 1 < x ≤ 5

6. A rectangular swimming pool is drawn on a coordinate plane with vertices at (0, 0), (20, 0), (20, 12), and (0, 12). A triangular diving area is marked off with vertices at (0, 0), (8, 0), and (0, 6). What is the area of the remaining pool surface that is available for swimming? Answer: 216 Solution: Length = 20 units, Width = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units The triangle has vertices at (0, 0), (8, 0), and (0, 6) This forms a right triangle with base = 8 units and height = 6 units Area of triangle = (1/2) × base × height = (1/2) × 8 × 6 = (1/2) × 48 =…
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular pool Length = 20 units, Width = 12 units Area of rectangle = length × width = 20 × 12 = 240 square units Step 2: Calculate the area of the triangular diving area The triangle has vertices at (0, 0), (8, 0), and (0, 6) This forms a right triangle with base = 8 units and height = 6 units Area of triangle = (1/2) × base × height = (1/2) × 8 × 6 = (1/2) × 48 = 24 square units Step 3: Calculate the remaining swimming area Remaining area = Total pool area - Diving area = 240 - 24 = 216 square units The answer is 216.

7. 3(2x - 5) + 7 ≥ 4(x + 3) - 1 and 2x - 7 < 9 = ? Answer: 5 ≤ x < 8 Solution: Solve the first inequality: 3(2x - 5) + 7 ≥ 4(x + 3) - 1 Distribute: 6x - 15 + 7 ≥ 4x + 12 - 1 Combine like terms: 6x - 8 ≥ 4x + 11 Subtract 4x from both sides: 2x - 8 ≥ 11 Add 8 to both sides: 2x ≥ 19 Divide by 2: x ≥ 9.5 Solve the second inequality: 2x - 7 < 9 Add 7 to both sides: 2x < 16…
   Full step-by-step solution

   Step 1: Solve the first inequality: 3(2x - 5) + 7 ≥ 4(x + 3) - 1 Step 2: Distribute: 6x - 15 + 7 ≥ 4x + 12 - 1 Step 3: Combine like terms: 6x - 8 ≥ 4x + 11 Step 4: Subtract 4x from both sides: 2x - 8 ≥ 11 Step 5: Add 8 to both sides: 2x ≥ 19 Step 6: Divide by 2: x ≥ 9.5 Step 7: Solve the second inequality: 2x - 7 < 9 Step 8: Add 7 to both sides: 2x < 16 Step 9: Divide by 2: x < 8 Step 10: Combine both solutions: x ≥ 9.5 and x < 8 Step 11: Since there is no number that is both ≥ 9.5 and < 8, there is no solution. The answer is no solution.

8. 3(x - 4) + 7 ≥ 16 and 2x + 5 < 25 Answer: 7 ≤ x < 10 Solution: 1) 3(x - 4) + 7 ≥ 16 2) 2x + 5 < 25 Solve the first inequality 3(x - 4) + 7 ≥ 16 First, distribute the 3: 3 * x - 3 * 4 + 7 ≥ 16 3x - 12 + 7 ≥ 16 3x - 5 ≥ 16 Add 5 to both sides: 3x ≥ 21 Divide both sides by 3: x ≥ 7 So from the first inequality: x ≥ 7.
   Full step-by-step solution

   Let's solve the compound inequality step by step. We have two inequalities: 1) 3(x - 4) + 7 ≥ 16 2) 2x + 5 < 25 --- **Step 1: Solve the first inequality 3(x - 4) + 7 ≥ 16** First, distribute the 3: 3 * x - 3 * 4 + 7 ≥ 16 3x - 12 + 7 ≥ 16 Combine like terms: 3x - 5 ≥ 16 Add 5 to both sides: 3x ≥ 21 Divide both sides by 3: x ≥ 7 So from the first inequality: x ≥ 7. --- **Step 2: Solve the second inequality 2x + 5 < 25** Subtract 5 from both sides: 2x < 20 Divide both sides by 2: x < 10 So from the second inequality: x < 10. --- **Step 3: Combine the solutions** From first inequality: x ≥ 7 From second inequality: x < 10 We need x to satisfy both conditions at the same time. So: 7 ≤ x < 10 --- **Final answer:** 7 ≤ x < 10