Linear Inequalities

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Answer Key & Explanations

Linear Inequalities · Grade 9 · Worksheet 2

1. 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

2. A tech startup is analyzing their server capacity needs. Their current user base grows at a rate where the number of users U(t) = 500 + 75t, where t is months from now. Their server capacity C can handle up to 2000 users simultaneously. The company wants to know when they'll need to upgrade their servers. Write an inequality that represents when the user base will exceed server capacity, then solve for t. Answer: t > 20 Solution: User function: U(t) = 500 + 75t Server capacity: C = 2000 We want to know when the user base exceeds capacity, so U(t) > C.
   Full step-by-step solution

   Step 1: Understand the problem We are given: User function: U(t) = 500 + 75t Server capacity: C = 2000 We want to know when the user base exceeds capacity, so U(t) > C. Step 2: Write the inequality Substitute U(t) and C into the inequality: 500 + 75t > 2000 Step 3: Isolate the term with t Subtract 500 from both sides: 75t > 2000 - 500 75t > 1500 Step 4: Solve for t Divide both sides by 75: t > 1500 / 75 Step 5: Simplify the division 1500 ÷ 75 = 20 So t > 20 Step 6: Interpret the result t > 20 means after 20 months from now, the user base will exceed the server capacity. Thus, they will need to upgrade their servers at t = 21 months if they plan ahead based on this model. Final answer: t > 20

3. Olivia is organizing a community art fair and needs to rent tables and chairs. Each table costs $15 to rent, and each chair costs $3 to rent. The venue requires that the number of chairs be at least three times the number of tables for proper seating. Olivia's total budget for renting tables and chairs is at most $189. Let t represent the number of tables. Write and solve a linear inequality to determine the maximum number of tables Olivia can rent. Answer: t ≤ 7 Solution: Let t be the number of tables. Since chairs must be at least three times the number of tables, the minimum number of chairs is 3t.
   Full step-by-step solution

   Step 1: Let t be the number of tables. Since chairs must be at least three times the number of tables, the minimum number of chairs is 3t. Step 2: Cost for tables: 15t Step 3: Cost for chairs: 3(3t) = 9t Step 4: Total cost inequality: 15t + 9t ≤ 189 Step 5: Combine like terms: 24t ≤ 189 Step 6: Divide both sides by 24: t ≤ 189/24 Step 7: Simplify: 189/24 = 63/8 = 7.875 Step 8: Since t must be a whole number (you can't rent a fraction of a table), the maximum integer value is t ≤ 7. The answer is t ≤ 7.

4. A tech company is developing a new smartphone app. The development cost is $15,000 and the company estimates they will earn $2.50 per download. The company wants to know how many downloads are needed to make a profit of at least $35,000. Write and solve an inequality to find the minimum number of downloads required, where d represents the number of downloads. Answer: 20000 Solution: - Development cost: $15,000 (this is a fixed cost they must cover before profit) - Earnings per download: $2.50 - Desired profit: at least $35,000 - Let \( d \) = number of downloads Total earnings from downloads = \( 2.50 \times d \) Profit = Total earnings − Development cost Profit = \( 2.50d…
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Understand the problem** - Development cost: $15,000 (this is a fixed cost they must cover before profit) - Earnings per download: $2.50 - Desired profit: at least $35,000 - Let \( d \) = number of downloads --- **Step 2: Write the expression for total earnings** Total earnings from downloads = \( 2.50 \times d \) --- **Step 3: Write the expression for profit** Profit = Total earnings − Development cost Profit = \( 2.50d - 15000 \) --- **Step 4: Set up the inequality** We want profit to be at least $35,000: \( 2.50d - 15000 \geq 35000 \) --- **Step 5: Solve the inequality** Add 15000 to both sides: \( 2.50d \geq 35000 + 15000 \) \( 2.50d \geq 50000 \) Divide both sides by 2.50: \( d \geq 50000 / 2.50 \) \( d \geq 20000 \) --- **Step 6: Interpret the result** The minimum number of downloads needed is 20,000. --- **Final answer:** 20000

5. 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.

6. Noah is a project manager at a software company. He needs to rent a conference room for a team training session. Room A charges a flat fee of $75 plus $15 per attendee. Room B charges a flat fee of $120 plus $9 per attendee. Noah's total budget for the room is at most $300. If x represents the number of attendees, determine the range of attendee numbers for which Room B is cheaper than or equal to Room A, while still staying within budget. Write and solve an inequality to find this range. Answer: x ≥ 8 and x ≤ 20 Solution: Write the cost expressions. Room A cost: 75 + 15x. Room B cost: 120 + 9x.
   Full step-by-step solution

   Step 1: Write the cost expressions. Room A cost: 75 + 15x. Room B cost: 120 + 9x. Step 2: Room B is cheaper or equal to Room A: 120 + 9x ≤ 75 + 15x Step 3: Subtract 9x from both sides: 120 ≤ 75 + 6x Step 4: Subtract 75 from both sides: 45 ≤ 6x Step 5: Divide both sides by 6: 7.5 ≤ x, which means x ≥ 7.5. Since x is number of attendees (a whole number), x ≥ 8. Step 6: Apply the budget constraint: both rooms must cost at most $300. Use Room B's cost since it is the cheaper option: 120 + 9x ≤ 300 Step 7: Subtract 120 from both sides: 9x ≤ 180 Step 8: Divide both sides by 9: x ≤ 20 Step 9: Combine the conditions: x ≥ 8 and x ≤ 20. So the range is 8 ≤ x ≤ 20.