Linear Inequalities

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Answer Key & Explanations

Linear Inequalities · Grade 9 · Worksheet 2

1. A tech company is analyzing their monthly revenue. Their basic subscription brings in $800 per month, and they earn an additional $12 for each premium user. The company needs their total monthly revenue to be at least $5000 to cover operating costs. If p represents the number of premium users, write and solve an inequality to determine the minimum number of premium users needed to meet this revenue requirement. Answer: 350 Solution: Write the expression for total revenue: 800 + 12p Set up the inequality: 800 + 12p ≥ 5000 Subtract 800 from both sides: 12p ≥ 4200 Divide both sides by 12: p ≥ 350 The minimum number of premium users needed is 350.
   Full step-by-step solution

   Step 1: Write the expression for total revenue: 800 + 12p Step 2: Set up the inequality: 800 + 12p ≥ 5000 Step 3: Subtract 800 from both sides: 12p ≥ 4200 Step 4: Divide both sides by 12: p ≥ 350 Step 5: The minimum number of premium users needed is 350.

2. Noah is managing a school fundraiser where he sells custom T-shirts. The fixed cost to set up the printing equipment is $150, and each T-shirt costs $7 to produce. Noah plans to sell each T-shirt for $15. If he wants to make a profit of at least $250, how many T-shirts must he sell? Write and solve an inequality to represent this situation, where x is the number of T-shirts sold. Answer: x ≥ 50 Solution: Write expressions for revenue and cost. Revenue from selling x T-shirts: 15x. Total cost: 150 (fixed) + 7x (variable).
   Full step-by-step solution

   Step 1: Write expressions for revenue and cost. Revenue from selling x T-shirts: 15x. Total cost: 150 (fixed) + 7x (variable). Step 2: Profit is revenue minus cost: 15x - (150 + 7x) = 15x - 150 - 7x = 8x - 150. Step 3: Noah wants profit at least $250, so 8x - 150 ≥ 250. Step 4: Add 150 to both sides: 8x ≥ 400. Step 5: Divide both sides by 8: x ≥ 50. Step 6: Check: If x = 50, profit = 8(50) - 150 = 400 - 150 = 250, which meets the requirement. So Noah must sell at least 50 T-shirts. The answer is x ≥ 50.

3. Tane is managing a community garden. He has 117 meters of fencing to enclose a rectangular plot where the length must be at least 9 meters more than three times the width. If the perimeter must use all the fencing, write and solve an inequality to find the possible widths of the garden that satisfy both conditions. Graph the solution on a number line. Answer: w ≤ 12.375 Solution: Let w represent the width in meters. The length L must be at least 9 more than three times the width: L ≥ 3w + 9. The perimeter of a rectangle is P = 2L + 2w.
   Full step-by-step solution

   Step 1: Let w represent the width in meters. The length L must be at least 9 more than three times the width: L ≥ 3w + 9. Step 2: The perimeter of a rectangle is P = 2L + 2w. Tane uses all 117 meters of fencing, so 2L + 2w = 117. Step 3: From the perimeter equation, solve for L: 2L = 117 − 2w, so L = (117 − 2w)/2 = 58.5 − w. Step 4: Now substitute this expression for L into the inequality L ≥ 3w + 9: 58.5 − w ≥ 3w + 9 Step 5: Subtract 9 from both sides: 49.5 − w ≥ 3w Step 6: Add w to both sides: 49.5 ≥ 4w Step 7: Divide both sides by 4: w ≤ 49.5/4 = 12.375 Step 8: Since width must be positive, the solution is 0 < w ≤ 12.375 meters. On a number line, draw an open circle at 0 (width positive) and a closed circle at 12.375, shading between them. The answer is w ≤ 12.375.

4. Aroha is organizing a school fundraiser. She has already collected $250 from sponsors. Each student who participates can raise an additional $15 by selling tickets. The school needs to raise at least $1,000 to fund the new library books. If x represents the number of students who participate, write and solve an inequality to find the minimum number of students needed to reach the fundraising goal. Answer: x ≥ 50 Solution: Aroha has already collected $250. Each student raises $15, so if x students participate, they raise 15x dollars. Total funds raised = 250 + 15x.
   Full step-by-step solution

   Step 1: Aroha has already collected $250. Step 2: Each student raises $15, so if x students participate, they raise 15x dollars. Step 3: Total funds raised = 250 + 15x. Step 4: The goal is at least $1,000, so we write the inequality: 250 + 15x ≥ 1000. Step 5: Subtract 250 from both sides: 15x ≥ 750. Step 6: Divide both sides by 15: x ≥ 50. Step 7: This means at least 50 students must participate. The answer is x ≥ 50.

5. A rectangular garden is designed with length (2x + 5) meters and width (x - 3) meters. The area of the garden must be at least 100 square meters. Write the inequality that represents this situation and solve for the possible values of x, assuming x > 3. Answer: x ≥ 7 Solution: When solving area inequalities involving polynomials, first expand the expression to form a quadratic inequality.
   Full step-by-step solution

   When solving area inequalities involving polynomials, first expand the expression to form a quadratic inequality. The solution involves finding where the quadratic expression meets or exceeds the given threshold value, considering the domain restrictions. The critical points help determine the valid intervals that satisfy the inequality condition.

6. 4(3x - 11) - 7(x + 2) ≥ 5x - 13 Answer: x ≤ -13 Solution: Distribute the coefficients: 4(3x - 11) becomes 12x - 44, and -7(x + 2) becomes -7x - 14. Combine like terms on the left side: 12x - 44 - 7x - 14 = 5x - 58. Write the inequality: 5x - 58 ≥ 5x - 13.
   Full step-by-step solution

   Step 1: Distribute the coefficients: 4(3x - 11) becomes 12x - 44, and -7(x + 2) becomes -7x - 14. Step 2: Combine like terms on the left side: 12x - 44 - 7x - 14 = 5x - 58. Step 3: Write the inequality: 5x - 58 ≥ 5x - 13. Step 4: Subtract 5x from both sides: -58 ≥ -13. Step 5: This is a false statement (-58 is not greater than or equal to -13). Therefore, there is no solution. The inequality has no solution.

7. 3(2x - 5) + 7 < 4x + 13 Answer: x < 10.5 Solution: 3(2x - 5) + 7 < 4x + 13 Distribute the 3 Multiply 3 by each term inside the parentheses: 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 < 13 2x - 8 < 13 Add 8 to both sides: 2x - 8 + 8 < 13 + 8 2x < 21 Divide both sides by…
   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** Multiply 3 by each term inside the parentheses: 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: Move the x terms to one side** Subtract 4x from both sides: 6x - 4x - 8 < 13 2x - 8 < 13 **Step 4: Move constant terms to the other side** 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 So the solution is: x < 10.5