Linear Inequalities

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Linear Inequalities · Grade 9 · Worksheet 3

1. A tech startup is analyzing their monthly server costs. Their basic hosting fee is $150 per month, plus $0.05 per gigabyte of data transferred. The company's budget for server costs cannot exceed $500 this month. Write and solve an inequality to determine the maximum gigabytes of data they can transfer while staying within budget. Answer: x ≤ 7000 Solution: Let \( x \) = the number of gigabytes of data transferred.
   Full step-by-step solution

   Let's break this problem down step by step. --- **Step 1: Define the variable** Let \( x \) = the number of gigabytes of data transferred. --- **Step 2: Write the cost expression** The cost has two parts: - Basic hosting fee: $150 - Cost per gigabyte: $0.05 per GB, so for \( x \) GB it costs \( 0.05 \times x \) Total cost = \( 150 + 0.05x \) --- **Step 3: Set up the inequality** The budget cannot exceed $500, so: \[ 150 + 0.05x \leq 500 \] --- **Step 4: Isolate the term with \( x \)** Subtract 150 from both sides: \[ 0.05x \leq 500 - 150 \] \[ 0.05x \leq 350 \] --- **Step 5: Solve for \( x \)** Divide both sides by 0.05: \[ x \leq \frac{350}{0.05} \] --- **Step 6: Perform the division** Dividing by 0.05 is the same as multiplying by 20 (since \( 1 / 0.05 = 20 \)): \[ x \leq 350 \times 20 \] \[ x \leq 7000 \] --- **Step 7: Interpret the result** The maximum gigabytes of data they can transfer while staying within budget is 7000 GB. --- **Final answer:** \[ x \leq 7000 \]

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

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

3. Noah is planning a fundraising event for his school's robotics club. He needs to rent a venue that costs a base fee of $150, plus $12 per attendee for catering. The club's total budget for the event cannot exceed $750. If x represents the number of attendees, write and solve an inequality to find the maximum number of attendees Noah can have while staying within the budget. Answer: x ≤ 50 Solution: Write the expression for total cost: 150 + 12x. The total cost cannot exceed $750, so set up the inequality: 150 + 12x ≤ 750. Subtract 150 from both sides: 12x ≤ 600.
   Full step-by-step solution

   Step 1: Write the expression for total cost: 150 + 12x. Step 2: The total cost cannot exceed $750, so set up the inequality: 150 + 12x ≤ 750. Step 3: Subtract 150 from both sides: 12x ≤ 600. Step 4: Divide both sides by 12: x ≤ 50. Step 5: Since x represents the number of attendees, it must be a whole number. The maximum number of attendees is 50. Final answer: x ≤ 50.

4. Hana is organizing a fundraising event for her school. She plans to sell tickets at $12 each, but the venue rental costs a flat fee of $240. If Hana wants to make a profit of at least $600, how many tickets must she sell? Write and solve an inequality to find the minimum number of tickets needed. Let t represent the number of tickets sold. Answer: t ≥ 70 Solution: Revenue from selling t tickets at $12 each is 12t dollars. The cost is $240. Profit = Revenue - Cost = 12t - 240.
   Full step-by-step solution

   Step 1: Revenue from selling t tickets at $12 each is 12t dollars. The cost is $240. Profit = Revenue - Cost = 12t - 240. Step 2: We want profit to be at least $600, so we write: 12t - 240 ≥ 600. Step 3: Add 240 to both sides: 12t ≥ 840. Step 4: Divide both sides by 12: t ≥ 70. Step 5: Since t represents the number of tickets, Hana must sell at least 70 tickets. The final answer is t ≥ 70.

5. A company manufactures phone cases and has a production cost of $2 per case plus $500 in fixed monthly costs. They sell each case for $7. To make a monthly profit of at least $1500, what is the minimum number of phone cases they must sell? Solve the inequality 7x - (2x + 500) ≥ 1500, where x represents the number of cases sold. Answer: 400 Solution: Production cost per case = $2 Fixed monthly cost = $500 Selling price per case = $7 Target monthly profit ≥ $1500 Revenue − Total Cost ≥ 1500 Revenue = 7x Total Cost = 2x + 500 7x − (2x + 500) ≥ 1500 7x − 2x − 500 ≥ 1500 5x − 500 ≥ 1500 Add 500 to both sides: 5x − 500 + 500 ≥ 1500 + 500 5x ≥…
   Full step-by-step solution

   Let's solve the inequality step by step. We are given: Production cost per case = $2 Fixed monthly cost = $500 Selling price per case = $7 Target monthly profit ≥ $1500 The profit equation is: Revenue − Total Cost ≥ 1500 Revenue = 7x Total Cost = 2x + 500 So: 7x − (2x + 500) ≥ 1500 --- **Step 1: Simplify the expression inside the inequality** 7x − 2x − 500 ≥ 1500 5x − 500 ≥ 1500 **Step 2: Isolate the term with x** Add 500 to both sides: 5x − 500 + 500 ≥ 1500 + 500 5x ≥ 2000 **Step 3: Solve for x** Divide both sides by 5: x ≥ 2000 / 5 x ≥ 400 --- **Step 4: Interpret the result** The smallest integer x satisfying the inequality is 400. Thus, the company must sell at least **400** phone cases to make a profit of at least $1500.

6. Mason is building a custom bookshelf. The design requires that the length of the shelf, in inches, is at least 7 more than twice its height. The total length of wood available for the entire shelf (the perimeter of the rectangular front face) cannot exceed 122 inches. Let h represent the height of the shelf in inches. Write and solve an inequality to find all possible heights for the shelf. Then graph the solution on a number line. Answer: h ≤ 18 Solution: Let h be the height. The length L is at least 7 more than twice the height, so L = 2h + 7. The perimeter P of a rectangle is given by P = 2L + 2h.
   Full step-by-step solution

   Step 1: Let h be the height. The length L is at least 7 more than twice the height, so L = 2h + 7. Step 2: The perimeter P of a rectangle is given by P = 2L + 2h. Step 3: Substitute L = 2h + 7 into the perimeter formula: P = 2(2h + 7) + 2h. Step 4: Simplify: P = 4h + 14 + 2h = 6h + 14. Step 5: The perimeter cannot exceed 122 inches, so: 6h + 14 ≤ 122. Step 6: Subtract 14 from both sides: 6h ≤ 108. Step 7: Divide both sides by 6: h ≤ 18. Step 8: Also, height must be positive: h > 0. Final answer: 0 < h ≤ 18. Graph: open circle at 0 with arrow to the right, closed circle at 18 with shading between them.

7. 3(2x - 4) - 5(x + 1) ≤ 2x - 7 Answer: x ≤ 10 Solution: Distribute the coefficients: 3(2x - 4) becomes 6x - 12, and -5(x + 1) becomes -5x - 5.
   Full step-by-step solution

   Step 1: Distribute the coefficients: 3(2x - 4) becomes 6x - 12, and -5(x + 1) becomes -5x - 5. So the inequality becomes: 6x - 12 - 5x - 5 ≤ 2x - 7 Step 2: Combine like terms on the left side: (6x - 5x) + (-12 - 5) = x - 17. So we have: x - 17 ≤ 2x - 7 Step 3: Subtract x from both sides: -17 ≤ x - 7 Step 4: Add 7 to both sides: -10 ≤ x Step 5: Rewrite the inequality: x ≥ -10 The solution is x ≥ -10.