Two-Step Inequalities

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Answer Key & Explanations

Two-Step Inequalities · Grade 7 · Worksheet 1

1. Sophia is organizing a charity bake sale. She already has $56 in donations and plans to sell cupcakes for $6 each. She needs at least $200 to meet her fundraising goal. Write and solve an inequality to determine the minimum number of cupcakes, c, she must sell. Answer: c ≥ 24 Solution: Let c represent the number of cupcakes sold. The total money is 56 + 6c. She needs at least $200, so the inequality is 56 + 6c ≥ 200.
   Full step-by-step solution

   Step 1: Let c represent the number of cupcakes sold. The total money is 56 + 6c. She needs at least $200, so the inequality is 56 + 6c ≥ 200. Step 2: Subtract 56 from both sides: 6c ≥ 144. Step 3: Divide both sides by 6: c ≥ 24. Step 4: The solution is c ≥ 24, meaning she must sell at least 24 cupcakes. Graph: Draw a number line, place a closed circle at 24, and shade to the right.

2. Sophia is organizing a school fundraiser and needs to order custom water bottles. The printing company charges a $36 setup fee plus $11 per water bottle. Sophia's budget for the water bottles is at most $586. Write and solve an inequality to determine the maximum number of water bottles, b, she can order without exceeding her budget. Answer: 50 Solution: Let b represent the number of water bottles. The total cost is the setup fee plus the cost per water bottle: 36 + 11b. The total cost must be at most $586, so write the inequality: 36 + 11b ≤ 586.
   Full step-by-step solution

   Step 1: Let b represent the number of water bottles. Step 2: The total cost is the setup fee plus the cost per water bottle: 36 + 11b. Step 3: The total cost must be at most $586, so write the inequality: 36 + 11b ≤ 586. Step 4: Subtract 36 from both sides to isolate the term with b: 11b ≤ 586 - 36, so 11b ≤ 550. Step 5: Divide both sides by 11 to solve for b: b ≤ 550 / 11, so b ≤ 50. Step 6: Since b must be a whole number, the maximum number of water bottles Sophia can order is 50. The answer is 50.

3. Emma is planning a school fundraiser by selling raffle tickets. She already has $150 raised from donations. Each raffle ticket is sold for $5. Emma's goal is to raise at least $400 in total. Write and solve an inequality to determine the minimum number of tickets, t, she needs to sell to meet her goal. Answer: t ≥ 50 Solution: Let t represent the number of tickets Emma sells. The money from ticket sales is 5t dollars. Added to the $150 she already has, the total is 150 + 5t.
   Full step-by-step solution

   Step 1: Let t represent the number of tickets Emma sells. Step 2: The money from ticket sales is 5t dollars. Added to the $150 she already has, the total is 150 + 5t. Step 3: She wants to raise at least $400, so the inequality is: 150 + 5t ≥ 400. Step 4: Subtract 150 from both sides: 5t ≥ 250. Step 5: Divide both sides by 5: t ≥ 50. Step 6: The solution means Emma must sell at least 50 tickets to reach her goal. On a number line, draw a closed circle at 50 and shade to the right. Answer: t ≥ 50.

4. Isabella is saving money to buy a new bicycle that costs $247. She already has $47 saved from her birthday. She plans to mow lawns in her neighborhood for $17 per lawn. What is the minimum number of lawns Isabella must mow to have enough money to buy the bicycle? Write and solve an inequality, then graph the solution on a number line. Answer: 12 Solution: Let l represent the number of lawns Isabella mows. She already has $47. She earns $17 per lawn, so her total money is 47 + 17l.
   Full step-by-step solution

   Step 1: Let l represent the number of lawns Isabella mows. Step 2: She already has $47. She earns $17 per lawn, so her total money is 47 + 17l. Step 3: She needs at least $247, so the inequality is: 47 + 17l >= 247 Step 4: Subtract 47 from both sides: 17l >= 200 Step 5: Divide both sides by 17: l >= 11.76... Step 6: Since she cannot mow a fraction of a lawn, the minimum number of whole lawns is 12. Step 7: To graph: Draw a number line from 0 to 15. Place a closed circle at 12 (because 12 is included). Shade the arrow to the right of 12, showing all numbers greater than or equal to 12. The answer is 12.

5. Tane is drawing a number line graph to solve the inequality 3x - 7 > 11. He draws a number line from -10 to 15, and places an open circle at the solution point. What number does the open circle sit on? Answer: 6 Solution: Start with the inequality 3x - 7 > 11. Add 7 to both sides to eliminate the constant: 3x - 7 + 7 > 11 + 7, which simplifies to 3x > 18. Divide both sides by 3 to solve for x: 3x / 3 > 18 / 3, which gives x > 6.
   Full step-by-step solution

   Step 1: Start with the inequality 3x - 7 > 11. Step 2: Add 7 to both sides to eliminate the constant: 3x - 7 + 7 > 11 + 7, which simplifies to 3x > 18. Step 3: Divide both sides by 3 to solve for x: 3x / 3 > 18 / 3, which gives x > 6. Step 4: On the number line, an open circle is placed at 6 to show that 6 is not included (since the inequality is strict 'greater than', not 'greater than or equal to'). The answer is 6.

6. Matiu is organizing a school fundraiser and needs to order custom hoodies. The printing company charges a flat setup fee of $120 plus $18 per hoodie. Matiu's budget for the hoodies is at most $600. Write and solve an inequality to determine the maximum number of hoodies, h, he can order without exceeding his budget. Then graph the solution on a number line. Answer: 26 Solution: Let h represent the number of hoodies. The total cost is the flat fee plus the cost per hoodie: 120 + 18h. The total cost must be at most the budget: 120 + 18h ≤ 600.
   Full step-by-step solution

   Step 1: Let h represent the number of hoodies. Step 2: The total cost is the flat fee plus the cost per hoodie: 120 + 18h. Step 3: The total cost must be at most the budget: 120 + 18h ≤ 600. Step 4: Subtract 120 from both sides: 18h ≤ 480. Step 5: Divide both sides by 18: h ≤ 26.67. Step 6: Since Matiu cannot order a fraction of a hoodie, the maximum whole number of hoodies is 26. Step 7: On a number line, draw a closed circle at 26 and shade to the left to show all values less than or equal to 26. The answer is 26.

7. 2(3x - 7) + 5 ≤ 4x + 15 Answer: x ≤ 12 Solution: Distribute the 2: 2(3x - 7) + 5 ≤ 4x + 15 becomes 6x - 14 + 5 ≤ 4x + 15 Combine like terms on the left: 6x - 9 ≤ 4x + 15 Subtract 4x from both sides: 6x - 9 - 4x ≤ 4x + 15 - 4x becomes 2x - 9 ≤ 15 Add 9 to both sides: 2x - 9 + 9 ≤ 15 + 9 becomes 2x ≤ 24 Divide both sides by 2: 2x/2 ≤ 24/2…
   Full step-by-step solution

   Step 1: Distribute the 2: 2(3x - 7) + 5 ≤ 4x + 15 becomes 6x - 14 + 5 ≤ 4x + 15 Step 2: Combine like terms on the left: 6x - 9 ≤ 4x + 15 Step 3: Subtract 4x from both sides: 6x - 9 - 4x ≤ 4x + 15 - 4x becomes 2x - 9 ≤ 15 Step 4: Add 9 to both sides: 2x - 9 + 9 ≤ 15 + 9 becomes 2x ≤ 24 Step 5: Divide both sides by 2: 2x/2 ≤ 24/2 becomes x ≤ 12 The solution is x ≤ 12.