Inequality Solutions

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Inequality Solutions · Grade 6 · Worksheet 2

1. Liam is organizing a school fundraiser and wants to raise at least $1200 for new library books. He has already collected $485 from local businesses. The student council can sell raffle tickets for $5 each. Write an inequality to determine how many raffle tickets, t, the students need to sell to reach their fundraising goal. Answer: t ≥ 143 Solution: When solving fundraising problems, you first determine how much more money is needed beyond what you already have. Then you divide that amount by the price per item to find the minimum number of items to sell.
   Full step-by-step solution

   When solving fundraising problems, you first determine how much more money is needed beyond what you already have. Then you divide that amount by the price per item to find the minimum number of items to sell. If the result isn't a whole number, you need to round up since you can't sell partial items.

2. Matiu is drawing a number line to represent the inequality x ≤ 1025. He marks a closed circle at 1025 and shades all the numbers to the left. Which of the following values are solutions to this inequality? Select all that apply: 1025, 1026, 1000, 1050, 1024. Answer: 1025, 1000, 1024 Solution: The inequality x ≤ 1025 means all numbers that are less than or equal to 1025. The closed circle at 1025 tells us that 1025 is included. The shading to the left means all numbers smaller than 1025 are also included.
   Full step-by-step solution

   Step 1: The inequality x ≤ 1025 means all numbers that are less than or equal to 1025. The closed circle at 1025 tells us that 1025 is included. The shading to the left means all numbers smaller than 1025 are also included. Step 2: Test each value: 1025 is equal to 1025, so it is a solution. 1026 is greater than 1025, so it is NOT a solution. 1000 is less than 1025, so it is a solution. 1050 is greater than 1025, so it is NOT a solution. 1024 is less than 1025, so it is a solution. Step 3: The solutions are 1025, 1000, and 1024.

3. Liam is organizing a school field trip to the science museum. The museum charges a flat fee of $200 for the group plus $12 per student. Liam's budget for the trip is $800. Write an inequality to determine the maximum number of students, s, that can attend while staying within the budget. Answer: s ≤ 50 Solution: Identify the fixed cost: $200 Identify the variable cost per student: $12 Write the expression for total cost: 200 + 12s Set up the inequality where total cost is less than or equal to the budget: 200 + 12s ≤ 800 Subtract 200 from both sides: 12s ≤ 600 Divide both sides by 12: s ≤ 50 The maximum…
   Full step-by-step solution

   Step 1: Identify the fixed cost: $200 Step 2: Identify the variable cost per student: $12 Step 3: Write the expression for total cost: 200 + 12s Step 4: Set up the inequality where total cost is less than or equal to the budget: 200 + 12s ≤ 800 Step 5: Subtract 200 from both sides: 12s ≤ 600 Step 6: Divide both sides by 12: s ≤ 50 Step 7: The maximum number of students that can attend is 50.

4. Is x = 12 a solution to x > 9? Answer: Yes Solution: The inequality is x > 9. This means any number greater than 9 is a solution. Test x = 12.
   Full step-by-step solution

   Step 1: The inequality is x > 9. This means any number greater than 9 is a solution. Step 2: Test x = 12. Is 12 > 9? Yes, because 12 is to the right of 9 on the number line. Step 3: Since 12 satisfies the inequality, x = 12 is a solution. The answer is Yes.

5. Hana is helping to organize a school science fair. She needs to set up display tables in the gymnasium. Each display table is 2 meters long, and the gymnasium is 30 meters long. The school rule says the total length of all tables cannot exceed the length of the gym. Hana has already placed 12 tables. Write an inequality to represent the number of additional tables, t, she can add without breaking the rule, and determine if she can add 4 more tables. Answer: No, 4 is not a solution to the inequality. Solution: Each table is 2 meters long. The gym is 30 meters long. The total length of all tables must be less than or equal to 30 meters.
   Full step-by-step solution

   Step 1: Each table is 2 meters long. The gym is 30 meters long. The total length of all tables must be less than or equal to 30 meters. Step 2: Hana has already placed 12 tables. The length of these tables is 12 * 2 = 24 meters. Step 3: Let t represent the number of additional tables. The length of additional tables is 2t meters. Step 4: The total length of all tables is 24 + 2t. The inequality is 24 + 2t ≤ 30. Step 5: Subtract 24 from both sides: 2t ≤ 6. Step 6: Divide both sides by 2: t ≤ 3. Step 7: The solution set is all values of t that are less than or equal to 3. On a number line, this would be a closed circle at 3 and an arrow pointing left. Step 8: To test if 4 is a solution, substitute t = 4 into the original inequality: 24 + 2(4) = 24 + 8 = 32. Since 32 ≤ 30 is false, 4 is not a solution. The answer is: No, 4 is not a solution to the inequality.

6. Mason is drawing a number line to represent the inequality x > -8. He marks a point at -8 and draws an arrow to the right. He then tests three numbers: -12, -5, and 0. Which of these numbers are solutions to the inequality x > -8? Explain your reasoning based on the number line. Answer: -5 and 0 are solutions; -12 is not a solution Solution: x > -8 means any number that is greater than -8. Since -12 < -8, it is NOT greater than -8. Since -5 > -8, it is a solution.
   Full step-by-step solution

   Step 1: Understand the inequality. x > -8 means any number that is greater than -8. On a number line, numbers increase as you move to the right. So the solution set includes all numbers to the right of -8 (but not -8 itself, since > means strictly greater). Step 2: Test each number: - Test -12: On the number line, -12 is to the left of -8. Since -12 < -8, it is NOT greater than -8. Therefore, -12 is not a solution. - Test -5: On the number line, -5 is to the right of -8. Since -5 > -8, it is a solution. - Test 0: On the number line, 0 is to the right of -8. Since 0 > -8, it is a solution. Step 3: Conclusion. The numbers -5 and 0 satisfy the inequality x > -8. The number -12 does not. Final answer: -5 and 0 are solutions; -12 is not a solution.

7. If 3x - 8 ≤ 25, what is the largest integer value of x? Answer: 11 Solution: Start with the inequality: 3x - 8 ≤ 25 Add 8 to both sides to isolate the term with x: 3x - 8 + 8 ≤ 25 + 8 Simplify: 3x ≤ 33 Divide both sides by 3 to solve for x: 3x ÷ 3 ≤ 33 ÷ 3 Simplify: x ≤ 11 The largest integer value that satisfies x ≤ 11 is 11.
   Full step-by-step solution

   Step 1: Start with the inequality: 3x - 8 ≤ 25 Step 2: Add 8 to both sides to isolate the term with x: 3x - 8 + 8 ≤ 25 + 8 Step 3: Simplify: 3x ≤ 33 Step 4: Divide both sides by 3 to solve for x: 3x ÷ 3 ≤ 33 ÷ 3 Step 5: Simplify: x ≤ 11 Step 6: The largest integer value that satisfies x ≤ 11 is 11. The answer is 11.