Inequality Solutions

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Inequality Solutions · Grade 6 · Worksheet 1

1. Emma is organizing a school fundraiser and needs to buy supplies. She has a budget of $350. The decorations cost $125, and she needs to buy snacks that cost $2.50 per person. If she expects between 80 and 100 attendees, write an inequality to determine how many people, p, can attend while staying within her budget. Answer: p ≤ 90 Solution: Identify the fixed cost: decorations = $125 Identify the variable cost: snacks = $2.50 per person Write the inequality for total cost: 125 + 2.50p ≤ 350 Subtract 125 from both sides: 2.50p ≤ 225 Divide both sides by 2.50: p ≤ 90 Check the range: Since she expects 80-100 attendees, the solution p…
   Full step-by-step solution

   Step 1: Identify the fixed cost: decorations = $125 Step 2: Identify the variable cost: snacks = $2.50 per person Step 3: Write the inequality for total cost: 125 + 2.50p ≤ 350 Step 4: Subtract 125 from both sides: 2.50p ≤ 225 Step 5: Divide both sides by 2.50: p ≤ 90 Step 6: Check the range: Since she expects 80-100 attendees, the solution p ≤ 90 fits within this range. The maximum number of people who can attend while staying within budget is 90.

2. If 3x - 15 ≤ 36, what is the largest integer value of x? Answer: 17 Solution: Start with the inequality: 3x - 15 ≤ 36 Add 15 to both sides: 3x - 15 + 15 ≤ 36 + 15 Simplify: 3x ≤ 51 Divide both sides by 3: 3x ÷ 3 ≤ 51 ÷ 3 Simplify: x ≤ 17 The largest integer value that satisfies x ≤ 17 is 17 The answer is 17.
   Full step-by-step solution

   Step 1: Start with the inequality: 3x - 15 ≤ 36 Step 2: Add 15 to both sides: 3x - 15 + 15 ≤ 36 + 15 Step 3: Simplify: 3x ≤ 51 Step 4: Divide both sides by 3: 3x ÷ 3 ≤ 51 ÷ 3 Step 5: Simplify: x ≤ 17 Step 6: The largest integer value that satisfies x ≤ 17 is 17 The answer is 17.

3. A school is planning a field trip and needs to rent buses. Each bus can hold 48 students. If there are 1250 students going on the trip, what is the minimum number of buses the school needs to rent so that all students have a seat? Write your answer as a whole number. Answer: 27 Solution: We have 1250 students, and each bus holds 48 students. We need to find the minimum number of buses so that all students have a seat. Divide the total number of students by the bus capacity.
   Full step-by-step solution

   Step 1: Understand the problem. We have 1250 students, and each bus holds 48 students. We need to find the minimum number of buses so that all students have a seat. Step 2: Divide the total number of students by the bus capacity. 1250 ÷ 48 = ? Step 3: Perform the division. 48 × 26 = 1248 1250 − 1248 = 2 So, 1250 ÷ 48 = 26 remainder 2. Step 4: Interpret the result. 26 buses would hold 1248 students, but we have 1250 students. That means 2 students would be left without a seat if we only use 26 buses. Step 5: Determine the minimum number of buses needed. Since we can’t leave any student behind, we need one more bus for the remaining 2 students. So, 26 + 1 = 27 buses. Step 6: Verify. 27 buses × 48 students = 1296 seats. 1296 ≥ 1250, so all students have a seat. Final Answer: 27

4. Hana is drawing a number line from -10 to 10 to represent the solution set for the inequality x > -3. She marks a point at -3 with an open circle and shades all the numbers to the right. Which of the following test values are solutions to the inequality x > -3: -5, 0, -3, 7, and -2? List all the solutions. Answer: 0, 7, -2 Solution: Shading to the right shows numbers larger than -3. - Is -5 > -3? Not a solution.
   Full step-by-step solution

   Step 1: Understand the inequality x > -3 means all numbers that are greater than -3. On the number line, an open circle at -3 shows -3 is not a solution. Shading to the right shows numbers larger than -3. Step 2: Test each value: - Is -5 > -3? No, -5 is less than -3 (to the left on the number line). Not a solution. - Is 0 > -3? Yes, 0 is greater than -3. Solution. - Is -3 > -3? No, -3 is equal to -3, not greater. Not a solution. - Is 7 > -3? Yes, 7 is greater than -3. Solution. - Is -2 > -3? Yes, -2 is greater than -3 (to the right on the number line). Solution. Step 3: The solutions are 0, 7, and -2. The answer is 0, 7, -2.

5. Olivia is drawing a number line to represent the solution set for the inequality x > 1253. She has already drawn a point at 1253. Should she draw an open circle or a closed circle at 1253, and in which direction should she shade the number line? Then, test whether x = 1253 is a solution to the inequality, and explain why or why not. Answer: Open circle at 1253, shade to the right; x = 1253 is NOT a solution because the inequality is strict (greater than, not greater than or equal to). Solution: The inequality x > 1253 means we want all numbers that are strictly greater than 1253. The number 1253 itself is not included because the symbol is '>' (greater than), not '≥' (greater than or equal to).
   Full step-by-step solution

   Step 1: Understand the inequality. The inequality x > 1253 means we want all numbers that are strictly greater than 1253. The number 1253 itself is not included because the symbol is '>' (greater than), not '≥' (greater than or equal to). Step 2: Draw the number line. Place a point at 1253. Since 1253 is not included in the solution set, use an open circle at 1253. Step 3: Determine the shading direction. Numbers greater than 1253 are to the right of 1253 on the number line. So, shade the number line to the right of the open circle. Step 4: Test x = 1253. Substitute 1253 into the inequality: 1253 > 1253. This statement is false because 1253 is equal to 1253, not greater than itself. Therefore, x = 1253 is NOT a solution. Final answer: Open circle at 1253, shade to the right; x = 1253 is not a solution.

6. Is x = 14 a solution to x > 9? Answer: Yes Solution: The inequality is x > 9. This means we are looking for values of x that are greater than 9. We need to test if x = 14 satisfies this condition.
   Full step-by-step solution

   Step 1: The inequality is x > 9. This means we are looking for values of x that are greater than 9. Step 2: We need to test if x = 14 satisfies this condition. Step 3: Compare 14 and 9. On a number line, 14 is to the right of 9, which means 14 is larger. Step 4: Since 14 > 9 is true, x = 14 is a solution to the inequality x > 9. The answer is Yes.

7. Noah 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. Noah's class has raised $650 from their bake sale. Write and solve an inequality to determine the maximum number of students, s, that can attend the field trip without exceeding their budget. Answer: 37 Solution: Write the inequality for the total cost. The museum charges $200 plus $12 per student: 200 + 12s Set up the inequality where the total cost is less than or equal to the budget: 200 + 12s ≤ 650 Subtract 200 from both sides: 12s ≤ 450 Divide both sides by 12: s ≤ 37.5 Since we can't have half a…
   Full step-by-step solution

   Step 1: Write the inequality for the total cost. The museum charges $200 plus $12 per student: 200 + 12s Step 2: Set up the inequality where the total cost is less than or equal to the budget: 200 + 12s ≤ 650 Step 3: Subtract 200 from both sides: 12s ≤ 450 Step 4: Divide both sides by 12: s ≤ 37.5 Step 5: Since we can't have half a student, the maximum number of whole students is 37. The answer is 37.

8. If 2x + 5 ≤ 17, what is the largest integer value of x? Answer: 6 Solution: Start with the inequality: 2x + 5 ≤ 17 Subtract 5 from both sides to isolate the term with x: 2x + 5 - 5 ≤ 17 - 5 Simplify: 2x ≤ 12 Divide both sides by 2 to solve for x: 2x / 2 ≤ 12 / 2 Simplify: x ≤ 6 The largest integer that is less than or equal to 6 is 6.
   Full step-by-step solution

   Step 1: Start with the inequality: 2x + 5 ≤ 17 Step 2: Subtract 5 from both sides to isolate the term with x: 2x + 5 - 5 ≤ 17 - 5 Step 3: Simplify: 2x ≤ 12 Step 4: Divide both sides by 2 to solve for x: 2x / 2 ≤ 12 / 2 Step 5: Simplify: x ≤ 6 Step 6: The largest integer that is less than or equal to 6 is 6. The answer is 6.