Compound Inequalities

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Answer Key & Explanations

Compound Inequalities · Grade 7 · Worksheet 1

1. 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

2. A rectangular garden is drawn on a coordinate grid with vertices at (0, 0), (18, 0), (18, 14), and (0, 14). A triangular flower bed is placed inside the garden with vertices at (0, 0), (10, 0), and (0, 7). The gardener wants to plant grass in the remaining area. However, a path for a fence must also be built along the inside of the garden such that the distance from the left and bottom edges is at least 2 meters and at most 5 meters. If the coordinates represent meters, write a compound inequality that describes the possible x-coordinates for the fence path, solve it, and graph the solution on a number line (describe the graph). Answer: 2 ≤ x ≤ 5 Solution: The distance from the left edge (x = 0) must be at least 2 meters, so x ≥ 2. The distance from the left edge must be at most 5 meters, so x ≤ 5.
   Full step-by-step solution

   Step 1: Understand the conditions. The distance from the left edge (x = 0) must be at least 2 meters, so x ≥ 2. The distance from the left edge must be at most 5 meters, so x ≤ 5. Step 2: Combine these using 'AND' because both conditions must be true at the same time: x ≥ 2 AND x ≤ 5. Step 3: Write as a compound inequality: 2 ≤ x ≤ 5. Step 4: Graph on a number line: Draw a number line from 0 to 6. Place a closed circle at 2 and a closed circle at 5. Shade the region between them. The answer is 2 ≤ x ≤ 5.

3. A factory produces between 12,500 and 15,800 units per day. Due to a new efficiency initiative, production is expected to increase by at least 2,300 units but no more than 3,700 units. Write a compound inequality that represents the possible daily production after the initiative, then solve for the range of possible production values. What is the minimum number of units the factory could produce after the initiative? Answer: 14800 Solution: The factory currently produces between 12,500 and 15,800 units per day. 12{,}500 \leq P \leq 15{,}800 The increase is at least 2,300 units but no more than 3,700 units.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the current production range** The factory currently produces between 12,500 and 15,800 units per day. So the current production \( P \) satisfies: \[ 12{,}500 \leq P \leq 15{,}800 \] --- **Step 2: Understand the increase** The increase is at least 2,300 units but no more than 3,700 units. So the increase \( I \) satisfies: \[ 2{,}300 \leq I \leq 3{,}700 \] --- **Step 3: Define new production after initiative** Let \( N \) = new production after initiative. Then \( N = P + I \). We want the possible range of \( N \). --- **Step 4: Find the minimum possible \( N \)** The smallest \( N \) occurs when \( P \) is smallest and \( I \) is smallest. Minimum \( P \) = 12,500 Minimum \( I \) = 2,300 So minimum \( N \) = 12,500 + 2,300 = 14,800. --- **Step 5: Find the maximum possible \( N \)** The largest \( N \) occurs when \( P \) is largest and \( I \) is largest. Maximum \( P \) = 15,800 Maximum \( I \) = 3,700 So maximum \( N \) = 15,800 + 3,700 = 19,500. --- **Step 6: Write the compound inequality for \( N \)** From the above, \( N \) ranges from 14,800 to 19,500: \[ 14{,}800 \leq N \leq 19{,}500 \] --- **Step 7: Answer the specific question** The problem asks: "What is the minimum number of units the factory could produce after the initiative?" From Step 4, that is 14,800. --- **Final Answer:** 14800

4. 2(4x - 6) + 10 ≥ 30 and 5x - 8 < 42 = ? Answer: 4 ≤ x < 10 Solution: Solve the first inequality: 2(4x - 6) + 10 ≥ 30 2(4x - 6) + 10 ≥ 30 8x - 12 + 10 ≥ 30 8x - 2 ≥ 30 8x ≥ 32 x ≥ 4 Solve the second inequality: 5x - 8 < 42 5x - 8 < 42 5x < 50 x < 10 Combine the solutions: x ≥ 4 AND x < 10 This means x is greater than or equal to 4, but less than 10.
   Full step-by-step solution

   Step 1: Solve the first inequality: 2(4x - 6) + 10 ≥ 30 2(4x - 6) + 10 ≥ 30 8x - 12 + 10 ≥ 30 8x - 2 ≥ 30 8x ≥ 32 x ≥ 4 Step 2: Solve the second inequality: 5x - 8 < 42 5x - 8 < 42 5x < 50 x < 10 Step 3: Combine the solutions: x ≥ 4 AND x < 10 This means x is greater than or equal to 4, but less than 10. Step 4: Write the compound inequality: 4 ≤ x < 10 The final answer is 4 ≤ x < 10.

5. A rectangular garden is drawn on a coordinate plane with vertices at (0, 0), (24, 0), (24, 16), and (0, 16). A triangular flower bed is planted with vertices at (0, 0), (9, 0), and (0, 12). What is the area of the remaining garden surface that is available for planting vegetables? Answer: 330 Solution: Calculate the area of the rectangular garden. Length = 24 units, Width = 16 units Area of rectangle = length × width = 24 × 16 = 384 square units Calculate the area of the triangular flower bed.
   Full step-by-step solution

   Step 1: Calculate the area of the rectangular garden. Length = 24 units, Width = 16 units Area of rectangle = length × width = 24 × 16 = 384 square units Step 2: Calculate the area of the triangular flower bed. The triangle has vertices at (0, 0), (9, 0), and (0, 12). This is a right triangle with base = 9 units (along the x-axis) and height = 12 units (along the y-axis). Area of triangle = (1/2) × base × height = (1/2) × 9 × 12 = (1/2) × 108 = 54 square units Step 3: Subtract the area of the flower bed from the total garden area. Remaining area = 384 - 54 = 330 square units The answer is 330.

6. 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

7. Emma is designing a science fair display board that must have an area between 1,200 and 1,800 square inches. The board's length is fixed at 40 inches. Write a compound inequality to represent all possible widths, w, in inches, that the board can have to meet the area requirements. Answer: 30 < w < 45 Solution: The area of a rectangle is calculated as length × width With length = 40 inches, the area is 40 × w The area must be between 1,200 and 1,800 square inches Write the compound inequality: 1,200 < 40w < 1,800 Divide all parts by 40 to solve for w: 1,200 ÷ 40 < w < 1,800 ÷ 40 Calculate: 30 < w < 45…
   Full step-by-step solution

   Step 1: The area of a rectangle is calculated as length × width Step 2: With length = 40 inches, the area is 40 × w Step 3: The area must be between 1,200 and 1,800 square inches Step 4: Write the compound inequality: 1,200 < 40w < 1,800 Step 5: Divide all parts by 40 to solve for w: 1,200 ÷ 40 < w < 1,800 ÷ 40 Step 6: Calculate: 30 < w < 45 Step 7: The width must be greater than 30 inches and less than 45 inches