Linear Inequalities

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Answer Key & Explanations

Linear Inequalities · Grade 9 · Worksheet 1

1. A right triangle is positioned on a coordinate plane with vertices at (0,0), (x,0), and (0,2x-4). The area of this triangle is greater than 12 square units. Write an inequality that represents all possible values of x, assuming x > 2. Answer: x > 4 Solution: Identify the base and height from the coordinates. The base is along the x-axis from (0,0) to (x,0), so base = x. The height is along the y-axis from (0,0) to (0,2x-4), so height = 2x - 4.
   Full step-by-step solution

   Step 1: Identify the base and height from the coordinates. The base is along the x-axis from (0,0) to (x,0), so base = x. The height is along the y-axis from (0,0) to (0,2x-4), so height = 2x - 4. Step 2: Write the area formula for a right triangle: Area = (1/2) * base * height = (1/2) * x * (2x - 4). Step 3: Set up the inequality for area > 12: (1/2) * x * (2x - 4) > 12. Step 4: Simplify the inequality: Multiply both sides by 2 to eliminate the fraction: x(2x - 4) > 24. Step 5: Expand: 2x^2 - 4x > 24. Step 6: Subtract 24 from both sides: 2x^2 - 4x - 24 > 0. Step 7: Divide the entire inequality by 2: x^2 - 2x - 12 > 0. Step 8: Factor the quadratic: (x - 4)(x + 2) > 0. Step 9: Find the critical points: x = 4 and x = -2. Step 10: Test intervals: Since x > 2 (given), we only consider x > 2. For x > 4, the product (x-4)(x+2) is positive. For 2 < x < 4, the product is negative. Step 11: Therefore, the solution is x > 4. The answer is x > 4.

2. 3(2x - 7) + 5 ≥ 2(4x - 3) - 9 Answer: x ≤ -2 Solution: Distribute the coefficients: 3(2x - 7) + 5 ≥ 2(4x - 3) - 9 becomes 6x - 21 + 5 ≥ 8x - 6 - 9 Combine like terms on each side: 6x - 16 ≥ 8x - 15 Subtract 6x from both sides: -16 ≥ 2x - 15 Add 15 to both sides: -1 ≥ 2x Divide both sides by 2: -1/2 ≥ x Rewrite the inequality: x ≤ -1/2 The solution…
   Full step-by-step solution

   Step 1: Distribute the coefficients: 3(2x - 7) + 5 ≥ 2(4x - 3) - 9 becomes 6x - 21 + 5 ≥ 8x - 6 - 9 Step 2: Combine like terms on each side: 6x - 16 ≥ 8x - 15 Step 3: Subtract 6x from both sides: -16 ≥ 2x - 15 Step 4: Add 15 to both sides: -1 ≥ 2x Step 5: Divide both sides by 2: -1/2 ≥ x Step 6: Rewrite the inequality: x ≤ -1/2 The solution is x ≤ -1/2.

3. Charlotte is a project manager at a construction firm. She is ordering steel beams for a new building. The supplier charges a flat delivery fee of $85, plus $12 for each steel beam ordered. Charlotte's budget for this order is at most $1,525. If x represents the number of steel beams she orders, write and solve an inequality to find the maximum number of steel beams Charlotte can order without exceeding her budget. Answer: x ≤ 120 Solution: The total cost is the flat delivery fee plus the cost per beam: 85 + 12x. This total cost must be at most the budget of $1,525, so the inequality is: 85 + 12x ≤ 1525. Subtract 85 from both sides: 12x ≤ 1440.
   Full step-by-step solution

   Step 1: The total cost is the flat delivery fee plus the cost per beam: 85 + 12x. Step 2: This total cost must be at most the budget of $1,525, so the inequality is: 85 + 12x ≤ 1525. Step 3: Subtract 85 from both sides: 12x ≤ 1440. Step 4: Divide both sides by 12: x ≤ 120. Step 5: Since x represents the number of steel beams, the maximum number Charlotte can order is 120. The answer is x ≤ 120.

4. Liam is designing a rectangular garden where the length must be at least 5 feet more than twice the width. If the perimeter of the garden cannot exceed 100 feet, and the width must be a positive value, what is the range of possible widths for Liam's garden? Write your answer using inequality notation. Answer: 0 < w ≤ 15 Solution: \( w \) = width of the garden (in feet) \( l \) = length of the garden (in feet) "Length must be at least 5 feet more than twice the width" means: l \geq 2w + 5 Perimeter \( P = 2l + 2w \) Perimeter cannot exceed 100 feet: 2l + 2w \leq 100 Divide through by 2: l + w \leq 50 From Step 2: \( l…
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let \( w \) = width of the garden (in feet) \( l \) = length of the garden (in feet) --- **Step 2: Translate the length condition** "Length must be at least 5 feet more than twice the width" means: \[ l \geq 2w + 5 \] --- **Step 3: Translate the perimeter condition** Perimeter \( P = 2l + 2w \) Perimeter cannot exceed 100 feet: \[ 2l + 2w \leq 100 \] Divide through by 2: \[ l + w \leq 50 \] --- **Step 4: Combine inequalities** From Step 2: \( l \geq 2w + 5 \) From Step 3: \( l \leq 50 - w \) So: \[ 2w + 5 \leq l \leq 50 - w \] For this to be possible, the lower bound must be less than or equal to the upper bound: \[ 2w + 5 \leq 50 - w \] --- **Step 5: Solve for \( w \)** \[ 2w + 5 \leq 50 - w \] \[ 2w + w \leq 50 - 5 \] \[ 3w \leq 45 \] \[ w \leq 15 \] --- **Step 6: Consider the width positivity condition** Width must be positive: \( w > 0 \) So combining: \[ 0 < w \leq 15 \] --- **Step 7: Check consistency** If \( w = 15 \), then \( l \geq 2(15) + 5 = 35 \) and \( l \leq 50 - 15 = 35 \), so \( l = 35 \) works. Perimeter = \( 2(35 + 15) = 100 \) (max allowed). If \( w \) is very small, say \( w = 1 \), then \( l \geq 7 \) and \( l \leq 49 \), so possible. --- **Final answer:** \[ 0 < w \leq 15 \]

5. Ava is a high school student organizing a fundraiser. She needs to rent a venue that costs a flat fee of $250, plus $15 per person for catering. Ava's total budget for the event is at most $1,000. If x represents the number of people attending, write and solve an inequality to find the maximum number of people she can invite while staying within budget. Answer: x ≤ 50 Solution: The total cost is the flat fee plus the per-person cost: $250 + 15x. This total must be at most $1,000, so the inequality is: 250 + 15x ≤ 1000. Subtract 250 from both sides: 15x ≤ 750.
   Full step-by-step solution

   Step 1: The total cost is the flat fee plus the per-person cost: $250 + 15x. Step 2: This total must be at most $1,000, so the inequality is: 250 + 15x ≤ 1000. Step 3: Subtract 250 from both sides: 15x ≤ 750. Step 4: Divide both sides by 15: x ≤ 50. Step 5: Since x represents the number of people, the maximum number of people Ava can invite is 50. The answer is x ≤ 50.

6. 3(2x - 4) - 2(x + 3) ≤ 5x - 7 Answer: x ≥ -5 Solution: Distribute the 3 and the 2: 3(2x - 4) - 2(x + 3) ≤ 5x - 7 becomes 6x - 12 - 2x - 6 ≤ 5x - 7 Combine like terms on the left side: 6x - 2x - 12 - 6 ≤ 5x - 7 becomes 4x - 18 ≤ 5x - 7 Subtract 4x from both sides: -18 ≤ x - 7 Add 7 to both sides: -11 ≤ x Rewrite the inequality: x ≥ -11 The solution…
   Full step-by-step solution

   Step 1: Distribute the 3 and the 2: 3(2x - 4) - 2(x + 3) ≤ 5x - 7 becomes 6x - 12 - 2x - 6 ≤ 5x - 7 Step 2: Combine like terms on the left side: 6x - 2x - 12 - 6 ≤ 5x - 7 becomes 4x - 18 ≤ 5x - 7 Step 3: Subtract 4x from both sides: -18 ≤ x - 7 Step 4: Add 7 to both sides: -11 ≤ x Step 5: Rewrite the inequality: x ≥ -11 The solution is x ≥ -11.

7. 3(2x - 5) + 7 > 4x + 13 Answer: x > 10.5 Solution: 3(2x - 5) + 7 > 4x + 13 Distribute the 3 3 * 2x = 6x 3 * (-5) = -15 6x - 15 + 7 > 4x + 13 -15 + 7 = -8 6x - 8 > 4x + 13 Subtract 4x from both sides: 6x - 4x - 8 > 4x - 4x + 13 2x - 8 > 13 Add 8 to both sides: 2x - 8 + 8 > 13 + 8 2x > 21 Divide both sides by 2: x > 21/2 x > 10.5 x > 10.5
   Full step-by-step solution

   Let's solve the inequality step by step. We start with: 3(2x - 5) + 7 > 4x + 13 **Step 1: Distribute the 3** 3 * 2x = 6x 3 * (-5) = -15 So we have: 6x - 15 + 7 > 4x + 13 **Step 2: Combine like terms on the left side** -15 + 7 = -8 So: 6x - 8 > 4x + 13 **Step 3: Get all x terms on one side** Subtract 4x from both sides: 6x - 4x - 8 > 4x - 4x + 13 2x - 8 > 13 **Step 4: Isolate the x term** Add 8 to both sides: 2x - 8 + 8 > 13 + 8 2x > 21 **Step 5: Solve for x** Divide both sides by 2: x > 21/2 x > 10.5 **Final answer:** x > 10.5

8. Solve the inequality and graph the solution on a number line: 4(3x - 7) - 9 ≥ 5(2x + 1) + 8 Answer: x ≥ 20 Solution: Distribute on both sides. Left side: 4(3x - 7) - 9 = 12x - 28 - 9 = 12x - 37 Right side: 5(2x + 1) + 8 = 10x + 5 + 8 = 10x + 13 Write the inequality. 12x - 37 ≥ 10x + 13 Subtract 10x from both sides.
   Full step-by-step solution

   Step 1: Distribute on both sides. Left side: 4(3x - 7) - 9 = 12x - 28 - 9 = 12x - 37 Right side: 5(2x + 1) + 8 = 10x + 5 + 8 = 10x + 13 Step 2: Write the inequality. 12x - 37 ≥ 10x + 13 Step 3: Subtract 10x from both sides. 12x - 10x - 37 ≥ 10x - 10x + 13 2x - 37 ≥ 13 Step 4: Add 37 to both sides. 2x - 37 + 37 ≥ 13 + 37 2x ≥ 40 Step 5: Divide both sides by 2. x ≥ 20 The solution is x ≥ 20. On a number line, draw a closed circle at 20 and shade to the right.