Graph Inequalities

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Answer Key & Explanations

Graph Inequalities · Grade 7 · Worksheet 2

1. Liam is designing a rectangular garden with a perimeter of at least 48 meters. The length of the garden must be 4 meters more than twice its width. Write an inequality that represents all possible widths for Liam's garden, then determine the minimum possible width in meters. Answer: 7 Solution: Let \( w \) = width of the garden in meters. Let \( l \) = length of the garden in meters.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let \( w \) = width of the garden in meters. Let \( l \) = length of the garden in meters. --- **Step 2: Translate the relationship between length and width** The problem says: "Length is 4 meters more than twice its width." So: \( l = 2w + 4 \) --- **Step 3: Write the perimeter formula** Perimeter \( P \) of a rectangle is: \( P = 2l + 2w \) --- **Step 4: Set up the inequality for perimeter** The perimeter is at least 48 meters: \( 2l + 2w \geq 48 \) --- **Step 5: Substitute \( l = 2w + 4 \) into the inequality** \( 2(2w + 4) + 2w \geq 48 \) --- **Step 6: Simplify** \( 4w + 8 + 2w \geq 48 \) \( 6w + 8 \geq 48 \) --- **Step 7: Solve for \( w \)** \( 6w \geq 48 - 8 \) \( 6w \geq 40 \) \( w \geq 40/6 \) \( w \geq 20/3 \) \( w \geq 6.666\ldots \) --- **Step 8: Interpret the result** Since \( w \) is a width in meters, and the inequality is \( w \geq 6.666\ldots \), the smallest possible width is the smallest number greater than or equal to \( 20/3 \) that makes sense in practical terms. But the problem likely expects an integer (since it says "minimum possible width in meters" and the answer is 7). So: \( w \) must be at least \( 20/3 \) meters ≈ 6.67 m. The smallest possible whole number width is 7 meters. --- **Final answer:** Minimum possible width = 7 meters.

2. Graph 3(2x - 9) + 5 > 4x - 7 Answer: x > 7.5 Solution: Distribute the 3: 3(2x - 9) = 6x - 27. The inequality becomes: 6x - 27 + 5 > 4x - 7. Combine like terms on the left: -27 + 5 = -22, so we have: 6x - 22 > 4x - 7.
   Full step-by-step solution

   Step 1: Distribute the 3: 3(2x - 9) = 6x - 27. The inequality becomes: 6x - 27 + 5 > 4x - 7. Step 2: Combine like terms on the left: -27 + 5 = -22, so we have: 6x - 22 > 4x - 7. Step 3: Subtract 4x from both sides: 6x - 4x - 22 > -7, which gives 2x - 22 > -7. Step 4: Add 22 to both sides: 2x > 15. Step 5: Divide both sides by 2: x > 7.5. The solution is x > 7.5. On a number line, draw an open circle at 7.5 and shade to the right.

3. Liam is designing a rectangular garden with a length of 12 meters. He wants the area of the garden to be at least 96 square meters. Write an inequality to represent all possible widths, w, that Liam can choose for his garden to meet this requirement. Answer: w ≥ 8 Solution: We have a rectangular garden with length 12 meters and width w meters. The area must be at least 96 square meters. Write the formula for the area of a rectangle.
   Full step-by-step solution

   Step 1: Understand the problem. We have a rectangular garden with length 12 meters and width w meters. The area must be at least 96 square meters. Step 2: Write the formula for the area of a rectangle. Area = length × width So, Area = 12 × w. Step 3: Set up the inequality for "area at least 96". "At least" means "greater than or equal to". So: 12 × w ≥ 96. Step 4: Solve the inequality for w. Divide both sides by 12: w ≥ 96 / 12. Step 5: Simplify the division. 96 ÷ 12 = 8. So: w ≥ 8. Step 6: Interpret the result. This means the width must be 8 meters or more to make the area at least 96 square meters. Final answer: w ≥ 8

4. 3(2x - 9) + 5 ≥ 7x - 11 Answer: x ≤ -11 Solution: Distribute the 3: 3(2x - 9) = 6x - 27. The inequality becomes: 6x - 27 + 5 ≥ 7x - 11. Combine like terms on the left: -27 + 5 = -22, so we have: 6x - 22 ≥ 7x - 11.
   Full step-by-step solution

   Step 1: Distribute the 3: 3(2x - 9) = 6x - 27. The inequality becomes: 6x - 27 + 5 ≥ 7x - 11. Step 2: Combine like terms on the left: -27 + 5 = -22, so we have: 6x - 22 ≥ 7x - 11. Step 3: Subtract 6x from both sides: 6x - 22 - 6x ≥ 7x - 11 - 6x → -22 ≥ x - 11. Step 4: Add 11 to both sides: -22 + 11 ≥ x - 11 + 11 → -11 ≥ x. Step 5: Rewrite the inequality with x on the left: x ≤ -11. The solution is x ≤ -11. On a number line, draw a closed circle at -11 and shade all numbers to the left.

5. Graph 4(2x - 6) ≥ 8x + 16 Answer: No solution (empty set) Solution: Distribute the 4 on the left side: 4(2x - 6) = 8x - 24. The inequality becomes 8x - 24 ≥ 8x + 16. Subtract 8x from both sides: 8x - 24 - 8x ≥ 8x + 16 - 8x, which simplifies to -24 ≥ 16.
   Full step-by-step solution

   Step 1: Distribute the 4 on the left side: 4(2x - 6) = 8x - 24. The inequality becomes 8x - 24 ≥ 8x + 16. Step 2: Subtract 8x from both sides: 8x - 24 - 8x ≥ 8x + 16 - 8x, which simplifies to -24 ≥ 16. Step 3: The statement -24 ≥ 16 is false because -24 is less than 16. Since the variable terms canceled out and the remaining statement is false, there is no value of x that makes the inequality true. Step 4: The solution is the empty set (no solution). On a number line, you would leave the line blank with no points shaded.

6. 2(3x - 5) + 4 ≥ 3(x + 2) - 1 Answer: x ≥ 3 Solution: When solving multi-step inequalities, you follow the same order of operations as with equations: distribute, combine like terms, then use inverse operations to isolate the variable.
   Full step-by-step solution

   When solving multi-step inequalities, you follow the same order of operations as with equations: distribute, combine like terms, then use inverse operations to isolate the variable. Remember that if you multiply or divide by a negative number, you must reverse the inequality symbol.

7. Emma is organizing a school fundraiser and needs to buy supplies. She has a budget of $500. She needs to buy at least 100 t-shirts that cost $4 each and some banners that cost $25 each. Write an inequality to represent the maximum number of banners she can buy, then solve for the actual maximum number of banners. Answer: 4 Solution: Calculate the cost of the required t-shirts: 100 t-shirts × $4 per t-shirt = $400 Subtract the t-shirt cost from the total budget: $500 - $400 = $100 remaining Divide the remaining money by the cost per banner: $100 ÷ $25 per banner = 4 banners Write the inequality: Let b represent the number of…
   Full step-by-step solution

   Step 1: Calculate the cost of the required t-shirts: 100 t-shirts × $4 per t-shirt = $400 Step 2: Subtract the t-shirt cost from the total budget: $500 - $400 = $100 remaining Step 3: Divide the remaining money by the cost per banner: $100 ÷ $25 per banner = 4 banners Step 4: Write the inequality: Let b represent the number of banners. The inequality is 4(100) + 25b ≤ 500 Step 5: Solve the inequality: 400 + 25b ≤ 500 → 25b ≤ 100 → b ≤ 4 The maximum number of banners Emma can buy is 4.

8. Liam is designing a rectangular garden with a length of 12 meters. The area of the garden must be at least 96 square meters. Write an inequality to represent the possible widths of the garden, then determine the minimum width that satisfies this condition. Answer: 8 meters Solution: We are told the garden is rectangular with length 12 meters. The area must be at least 96 square meters. Write the area formula for a rectangle.
   Full step-by-step solution

   We are told the garden is rectangular with length 12 meters. The area must be at least 96 square meters. Step 1: Write the area formula for a rectangle. Area = length × width Let width = w meters. So Area = 12 × w. Step 2: The area must be at least 96, so: 12 × w ≥ 96. Step 3: Solve the inequality for w. Divide both sides by 12: w ≥ 96 / 12. Step 4: Calculate 96 / 12. 96 ÷ 12 = 8. Step 5: Conclusion. w ≥ 8. This means the width must be at least 8 meters. The minimum width that satisfies the condition is 8 meters. ANSWER: 8 meters