Two-Step Inequalities

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Answer Key & Explanations

Two-Step Inequalities · Grade 7 · Worksheet 3

1. Emma is organizing a bake sale for her school's sports team. She needs to raise at least $500 to buy new equipment. She already has $125 from a previous fundraiser. If each batch of cookies she bakes earns $15, write and solve an inequality to find the minimum number of batches of cookies Emma must bake to reach her goal. Then graph the solution on a number line. Answer: 25 Solution: Let b represent the number of batches of cookies Emma needs to bake. The total money raised is the $125 she already has plus $15 for each batch, or 125 + 15b.
   Full step-by-step solution

   Step 1: Let b represent the number of batches of cookies Emma needs to bake. Step 2: The total money raised is the $125 she already has plus $15 for each batch, or 125 + 15b. Step 3: She needs at least $500, so the inequality is: 125 + 15b ≥ 500. Step 4: Subtract 125 from both sides to isolate the term with b: 125 + 15b - 125 ≥ 500 - 125, which simplifies to 15b ≥ 375. Step 5: Divide both sides by 15 to solve for b: 15b / 15 ≥ 375 / 15, which gives b ≥ 25. Step 6: This means Emma must bake at least 25 batches of cookies. Graph: On a number line, draw a closed circle at 25 (because 25 is included) and shade the number line to the right of 25 to show all numbers greater than or equal to 25. The answer is 25.

2. Liam is saving money to buy a new video game that costs $65. He already has $20 saved from his allowance. Each week, he earns an additional $15 from doing chores. Write and solve an inequality to determine the minimum number of weeks Liam needs to work to afford the game. Answer: 3 Solution: Liam wants a video game that costs $65. He already has $20 saved. Each week he earns $15 from chores.
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Understand the problem** Liam wants a video game that costs $65. He already has $20 saved. Each week he earns $15 from chores. We need to find the minimum number of weeks \( w \) so that his total savings is at least $65. --- **Step 2: Write an expression for total money after \( w \) weeks** Money from allowance already saved = $20 Money from chores in \( w \) weeks = \( 15 \times w \) Total money = \( 20 + 15w \) --- **Step 3: Set up the inequality** He needs at least $65, so: \[ 20 + 15w \geq 65 \] --- **Step 4: Solve the inequality** Subtract 20 from both sides: \[ 15w \geq 65 - 20 \] \[ 15w \geq 45 \] Divide both sides by 15: \[ w \geq 45 / 15 \] \[ w \geq 3 \] --- **Step 5: Interpret the result** \( w \geq 3 \) means the smallest whole number of weeks is 3. Check: After 3 weeks, total = \( 20 + 15 \times 3 = 20 + 45 = 65 \), which is exactly enough. --- **Final answer:** 3 weeks

3. A rectangular garden is drawn on a coordinate plane with vertices at (2, 1), (8, 1), (8, 5), and (2, 5). The gardener wants to plant flowers in the region where y > 2x - 7. Determine which vertices of the rectangle satisfy this inequality. Answer: (8,1), (8,5), (2,5) Solution: To solve problems like this, you need to understand how to test points in linear inequalities. For any point (x, y), you substitute these values into the inequality and check if the resulting statement is true.
   Full step-by-step solution

   To solve problems like this, you need to understand how to test points in linear inequalities. For any point (x, y), you substitute these values into the inequality and check if the resulting statement is true. Points that satisfy the inequality will be located on one side of the boundary line formed by the corresponding equation. In geometric terms, you're determining which corners of a shape fall within a specific region defined by the inequality.

4. 3(x - 4) + 7 ≤ 25 Answer: x ≤ 10 Solution: Distribute the 3: 3(x - 4) + 7 ≤ 25 becomes 3x - 12 + 7 ≤ 25 Combine like terms: 3x - 5 ≤ 25 Add 5 to both sides: 3x ≤ 30 Divide both sides by 3: x ≤ 10 The solution is x ≤ 10.
   Full step-by-step solution

   Step 1: Distribute the 3: 3(x - 4) + 7 ≤ 25 becomes 3x - 12 + 7 ≤ 25 Step 2: Combine like terms: 3x - 5 ≤ 25 Step 3: Add 5 to both sides: 3x ≤ 30 Step 4: Divide both sides by 3: x ≤ 10 The solution is x ≤ 10.

5. 7x - 12 > 37. Solve and graph. Answer: x > 7 Solution: Add 12 to both sides to isolate the term with x: 7x - 12 + 12 > 37 + 12 becomes 7x > 49. Divide both sides by 7 (positive, so inequality direction stays the same): 7x/7 > 49/7 becomes x > 7. The solution is x > 7.
   Full step-by-step solution

   Step 1: Add 12 to both sides to isolate the term with x: 7x - 12 + 12 > 37 + 12 becomes 7x > 49. Step 2: Divide both sides by 7 (positive, so inequality direction stays the same): 7x/7 > 49/7 becomes x > 7. The solution is x > 7. On a number line, draw an open circle at 7 and shade to the right.

6. Aisha is planning a school field trip to a science museum. The museum charges a flat fee of $250 for the group plus $12 per student. The school has budgeted at most $1000 for this trip. Write and solve an inequality to determine the maximum number of students who can attend the field trip while staying within the budget. Answer: 62 Solution: Let x represent the number of students attending the field trip. The total cost is the flat fee plus the cost per student: 250 + 12x The school can spend at most $1000, so we write: 250 + 12x ≤ 1000 Subtract 250 from both sides: 12x ≤ 750 Divide both sides by 12: x ≤ 62.5 Since we can't have…
   Full step-by-step solution

   Step 1: Let x represent the number of students attending the field trip. Step 2: The total cost is the flat fee plus the cost per student: 250 + 12x Step 3: The school can spend at most $1000, so we write: 250 + 12x ≤ 1000 Step 4: Subtract 250 from both sides: 12x ≤ 750 Step 5: Divide both sides by 12: x ≤ 62.5 Step 6: Since we can't have half a student, we round down to the nearest whole number: x ≤ 62 The maximum number of students who can attend is 62.

7. 2(x - 5) + 7 ≤ 15 Answer: x ≤ 9 Solution: Distribute the 2: 2(x - 5) + 7 ≤ 15 becomes 2x - 10 + 7 ≤ 15 Combine like terms: 2x - 3 ≤ 15 Add 3 to both sides: 2x ≤ 18 Divide both sides by 2: x ≤ 9 The solution is x ≤ 9.
   Full step-by-step solution

   Step 1: Distribute the 2: 2(x - 5) + 7 ≤ 15 becomes 2x - 10 + 7 ≤ 15 Step 2: Combine like terms: 2x - 3 ≤ 15 Step 3: Add 3 to both sides: 2x ≤ 18 Step 4: Divide both sides by 2: x ≤ 9 The solution is x ≤ 9.

8. Matiu is drawing a number line graph to represent the solution to the inequality 4x - 18 > 50. He has drawn a number line from 0 to 30. What number should he put an open circle on, and which direction should he shade to correctly graph the solution? Answer: Open circle at 17, shade to the right Solution: Solve the inequality 4x - 18 > 50 Add 18 to both sides: 4x - 18 + 18 > 50 + 18 4x > 68 Divide both sides by 4: 4x/4 > 68/4 x > 17 Since the inequality is x > 17, we use an open circle at 17 (because 17 is not included).
   Full step-by-step solution

   Step 1: Solve the inequality 4x - 18 > 50 Add 18 to both sides: 4x - 18 + 18 > 50 + 18 4x > 68 Divide both sides by 4: 4x/4 > 68/4 x > 17 Step 2: Interpret the solution for graphing Since the inequality is x > 17, we use an open circle at 17 (because 17 is not included). The solution is all numbers greater than 17, so we shade to the right of 17. The answer is: open circle at 17, shade to the right.