Create Inequalities

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Answer Key & Explanations

Create Inequalities · Grade 9 · Worksheet 3

1. Aisha is designing a rectangular parking lot for a new shopping center. The length of the lot must be at least 10 meters more than three times its width. Due to zoning regulations, the area of the parking lot cannot exceed 4000 square meters. If w represents the width of the parking lot in meters, write an inequality that represents all possible widths that satisfy these conditions. Answer: 0 < w ≤ 25 Solution: When working with area constraints involving rectangles where length is expressed in terms of width, we create a quadratic inequality.
   Full step-by-step solution

   When working with area constraints involving rectangles where length is expressed in terms of width, we create a quadratic inequality. The solution involves finding where the product of the dimensions satisfies the maximum area requirement while ensuring all measurements remain positive. This type of problem appears in various design and planning scenarios where space limitations must be considered.

2. Mason is saving to buy a new video game console that costs $427. He already has $127 and plans to save $37 each week from his allowance. What is the minimum number of weeks he needs to save to afford the console? Write an inequality and solve. Answer: w ≥ 9 Solution: Let w represent the number of weeks Mason saves. He already has $127 and saves $37 per week, so total saved = 127 + 37w. He needs at least $427, so the inequality is 127 + 37w ≥ 427.
   Full step-by-step solution

   Step 1: Let w represent the number of weeks Mason saves. Step 2: He already has $127 and saves $37 per week, so total saved = 127 + 37w. Step 3: He needs at least $427, so the inequality is 127 + 37w ≥ 427. Step 4: Subtract 127 from both sides: 37w ≥ 300. Step 5: Divide both sides by 37: w ≥ 300/37 ≈ 8.108. Step 6: Since w must be a whole number, the minimum number of weeks is 9. Final answer: w ≥ 9.

3. Emma is saving to buy a new bicycle that costs $455. She already has $95 and plans to save $45 each week from her allowance. Write an inequality to represent the minimum number of weeks, w, she needs to save to afford the bicycle, and solve for w. Answer: w ≥ 8 Solution: Let w represent the number of weeks Emma saves. She already has $95 and saves $45 per week, so total saved = 95 + 45w. She needs at least $455, so the inequality is 95 + 45w ≥ 455.
   Full step-by-step solution

   Step 1: Let w represent the number of weeks Emma saves. Step 2: She already has $95 and saves $45 per week, so total saved = 95 + 45w. Step 3: She needs at least $455, so the inequality is 95 + 45w ≥ 455. Step 4: Subtract 95 from both sides: 45w ≥ 360. Step 5: Divide both sides by 45: w ≥ 8. Step 6: Since w represents whole weeks, the minimum number of weeks is 8. Final answer: w ≥ 8.

4. Liam is designing a rectangular garden with a perimeter of 40 meters. He wants the length of the garden to be at least 5 meters more than twice its width. Write an inequality that represents all possible widths (in meters) that satisfy Liam's design requirements. Answer: w ≤ 5 Solution: - \( w \) = width of the garden (in meters) - \( l \) = length of the garden (in meters) Perimeter of a rectangle = \( 2l + 2w \) Given: \( 2l + 2w = 40 \) Divide both sides by 2: \( l + w = 20 \) \( l = 20 - w \) "Length is at least 5 meters more than twice its width" means: \( l \geq 2w + 5 \)…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let - \( w \) = width of the garden (in meters) - \( l \) = length of the garden (in meters) --- **Step 2: Use the perimeter information** Perimeter of a rectangle = \( 2l + 2w \) Given: \( 2l + 2w = 40 \) Divide both sides by 2: \( l + w = 20 \) So: \( l = 20 - w \) --- **Step 3: Translate the condition on length** "Length is at least 5 meters more than twice its width" means: \( l \geq 2w + 5 \) --- **Step 4: Substitute \( l = 20 - w \) into the inequality** \( 20 - w \geq 2w + 5 \) --- **Step 5: Solve for \( w \)** \( 20 - w - 2w \geq 5 \) \( 20 - 3w \geq 5 \) Subtract 20 from both sides: \( -3w \geq -15 \) Divide by -3 (remember to reverse the inequality sign): \( w \leq 5 \) --- **Step 6: Consider physical constraints** Width must be positive: \( w > 0 \) So the final possible widths are: \( 0 < w \leq 5 \) But since the problem asks for the inequality representing all possible widths satisfying the design requirements, the key inequality is: \( w \leq 5 \) --- **Final answer:** w ≤ 5

5. Emma is designing a rectangular banner for a school event. The banner's length must be at least 3 meters more than twice its width. If the area of the banner must be at least 54 square meters, and the width is represented by w meters, write and solve an inequality to determine the possible width values for Emma's banner. Answer: w ≥ 4.5 Solution: Write the relationship between length and width. Length = 2w + 3 Write the area formula: Area = length × width = (2w + 3) × w Set up the inequality: (2w + 3) × w ≥ 54 Expand: 2w² + 3w ≥ 54 Subtract 54 from both sides: 2w² + 3w - 54 ≥ 0 Solve the quadratic equation 2w² + 3w - 54 = 0 Use the…
   Full step-by-step solution

   Step 1: Write the relationship between length and width. Length = 2w + 3 Step 2: Write the area formula: Area = length × width = (2w + 3) × w Step 3: Set up the inequality: (2w + 3) × w ≥ 54 Step 4: Expand: 2w² + 3w ≥ 54 Step 5: Subtract 54 from both sides: 2w² + 3w - 54 ≥ 0 Step 6: Solve the quadratic equation 2w² + 3w - 54 = 0 Step 7: Use the quadratic formula: w = [-3 ± √(9 + 432)] / 4 = [-3 ± √441] / 4 = [-3 ± 21] / 4 Step 8: Calculate the two solutions: w = (-3 + 21)/4 = 18/4 = 4.5 and w = (-3 - 21)/4 = -24/4 = -6 Step 9: Since width must be positive, we discard w = -6 Step 10: Test intervals: For w < 4.5, the expression is negative; for w > 4.5, the expression is positive Step 11: Therefore, the solution is w ≥ 4.5 The answer is w ≥ 4.5.

6. Aisha is designing a rectangular banner for a school event. The banner must have an area of at least 120 square feet. If the length of the banner is 4 feet more than its width, write and solve an inequality to determine the minimum possible width of the banner in feet. Answer: w ≥ 10 Solution: When solving area-based inequality problems, we express the area in terms of one variable using the given relationship between length and width.
   Full step-by-step solution

   When solving area-based inequality problems, we express the area in terms of one variable using the given relationship between length and width. The resulting quadratic inequality can be solved by finding where the quadratic expression equals the required area, then determining which interval satisfies the inequality based on the parabola's shape and the physical constraints of the problem (like positive dimensions).

7. Noah is saving to buy a new gaming console that costs $680. He already has $200 and plans to save $30 each week from his allowance. What is the minimum number of weeks he needs to save to afford the console? Write an inequality and solve for the number of weeks. Answer: w ≥ 16 Solution: Let w represent the number of weeks Noah saves. He already has $200 and saves $30 per week, so total saved = 200 + 30w. He needs at least $680, so the inequality is 200 + 30w ≥ 680.
   Full step-by-step solution

   Step 1: Let w represent the number of weeks Noah saves. Step 2: He already has $200 and saves $30 per week, so total saved = 200 + 30w. Step 3: He needs at least $680, so the inequality is 200 + 30w ≥ 680. Step 4: Subtract 200 from both sides: 30w ≥ 480. Step 5: Divide both sides by 30: w ≥ 16. Step 6: Since w must be a whole number, the minimum number of weeks is 16. Final answer: w ≥ 16.