Graph Inequalities

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Answer Key & Explanations

Graph Inequalities · Grade 7 · Worksheet 3

1. Graph 3(2x - 7) ≥ 5x + 12 Answer: x ≥ 33 Solution: Distribute the 3 on the left side: 3(2x - 7) = 6x - 21. The inequality becomes: 6x - 21 ≥ 5x + 12. Subtract 5x from both sides to get variable terms on the left: 6x - 21 - 5x ≥ 5x + 12 - 5x → x - 21 ≥ 12.
   Full step-by-step solution

   Step 1: Distribute the 3 on the left side: 3(2x - 7) = 6x - 21. The inequality becomes: 6x - 21 ≥ 5x + 12. Step 2: Subtract 5x from both sides to get variable terms on the left: 6x - 21 - 5x ≥ 5x + 12 - 5x → x - 21 ≥ 12. Step 3: Add 21 to both sides to isolate x: x - 21 + 21 ≥ 12 + 21 → x ≥ 33. Step 4: Graph the solution on a number line: Draw a number line, place a closed circle at 33 (because the inequality includes 'equal to'), and shade the arrow to the right (toward larger numbers) to show all values greater than or equal to 33. The solution is x ≥ 33.

2. 3(2x - 5) ≥ 4x + 7 Answer: x ≥ 11 Solution: 3(2x - 5) ≥ 4x + 7 Distribute the 3 Multiply 3 by each term inside the parentheses: 3 * 2x = 6x 3 * (-5) = -15 6x - 15 ≥ 4x + 7 Subtract 4x from both sides: 6x - 4x - 15 ≥ 4x - 4x + 7 2x - 15 ≥ 7 Add 15 to both sides: 2x - 15 + 15 ≥ 7 + 15 2x ≥ 22 Divide both sides by 2: x ≥ 11 The solution…
   Full step-by-step solution

   Let's solve the inequality step by step. We start with: 3(2x - 5) ≥ 4x + 7 **Step 1: Distribute the 3** Multiply 3 by each term inside the parentheses: 3 * 2x = 6x 3 * (-5) = -15 So we get: 6x - 15 ≥ 4x + 7 **Step 2: Move all terms with x to one side** Subtract 4x from both sides: 6x - 4x - 15 ≥ 4x - 4x + 7 2x - 15 ≥ 7 **Step 3: Move constant terms to the other side** Add 15 to both sides: 2x - 15 + 15 ≥ 7 + 15 2x ≥ 22 **Step 4: Solve for x** Divide both sides by 2: x ≥ 11 **Step 5: Interpret the result** The solution means x can be any number greater than or equal to 11. **Final answer:** x ≥ 11

3. Mere is buying supplies for a school project. Each package of markers costs $8. Mere has a budget of $50 to spend on markers. What is the maximum number of packages Mere can buy? Graph the solution on a number line. Answer: 6 Solution: Let p = number of packages. Inequality: 8p ≤ 50. Solve: p ≤ 50 / 8 = 6.
   Full step-by-step solution

   Step 1: Let p = number of packages. Inequality: 8p ≤ 50. Step 2: Solve: p ≤ 50 / 8 = 6. Step 3: Since p must be a whole number, the maximum is 6. Step 4: Graph: closed circle at 6, arrow to the left.

4. Emma is organizing a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $150 plus $8 per shirt. Emma's budget for t-shirts is at most $1,000. Write an inequality to represent this situation, then determine the maximum number of t-shirts she can order without exceeding her budget. Answer: 106 Solution: Let x represent the number of t-shirts Emma can order. The total cost is the setup fee plus the cost per shirt: 150 + 8x The budget constraint means the total cost must be less than or equal to $1,000: 150 + 8x ≤ 1000 Subtract 150 from both sides: 8x ≤ 850 Divide both sides by 8: x ≤ 106.25…
   Full step-by-step solution

   Step 1: Let x represent the number of t-shirts Emma can order. Step 2: The total cost is the setup fee plus the cost per shirt: 150 + 8x Step 3: The budget constraint means the total cost must be less than or equal to $1,000: 150 + 8x ≤ 1000 Step 4: Subtract 150 from both sides: 8x ≤ 850 Step 5: Divide both sides by 8: x ≤ 106.25 Step 6: Since Emma can't order a fraction of a shirt, the maximum number is 106. The answer is 106.

5. Hana is tracking the temperature of a chemical solution for a science experiment. The solution must be kept below 14 degrees Celsius. The current temperature is 2 degrees Celsius, and the temperature increases by 4 degrees every hour. Graph the inequality that represents the number of hours (h) that can pass before the solution exceeds the safe temperature. Use a number line from 0 to 5. Answer: An open circle at 3 with an arrow to the left. Solution: Write the inequality. The temperature starts at 2 and increases by 4 each hour, so after h hours, temperature = 2 + 4h. It must stay below 14: 2 + 4h < 14.
   Full step-by-step solution

   Step 1: Write the inequality. The temperature starts at 2 and increases by 4 each hour, so after h hours, temperature = 2 + 4h. It must stay below 14: 2 + 4h < 14. Step 2: Solve the inequality. 2 + 4h < 14 Subtract 2 from both sides: 4h < 12 Divide both sides by 4: h < 3 Step 3: Interpret the solution. h < 3 means any number of hours less than 3. Since time is continuous (not just whole hours), the solution includes values like 2.5 hours. Step 4: Graph on a number line from 0 to 5. Place an open circle at 3 (since 3 is not included, because h must be strictly less than 3). Draw an arrow pointing to the left from the open circle, covering all numbers less than 3. Final answer: An open circle at 3 with an arrow to the left.

6. Aroha is graphing the solution to the inequality 3x - 5 > 7 on a number line. First, solve the inequality. Then, describe what the graph will look like: What type of circle (open or closed) will be at the boundary point, and in which direction (left or right) will the ray be shaded? Answer: Open circle at 4, shaded to the right Solution: Solve the inequality 3x - 5 > 7. Add 5 to both sides: 3x - 5 + 5 > 7 + 5, which simplifies to 3x > 12. Divide both sides by 3: 3x / 3 > 12 / 3, which gives x > 4.
   Full step-by-step solution

   Step 1: Solve the inequality 3x - 5 > 7. Add 5 to both sides: 3x - 5 + 5 > 7 + 5, which simplifies to 3x > 12. Divide both sides by 3: 3x / 3 > 12 / 3, which gives x > 4. Step 2: Interpret the solution. x > 4 means all numbers greater than 4, but not including 4 itself. Step 3: Determine the circle type. Since the inequality is strictly greater than (not greater than or equal to), we use an open circle at 4. Step 4: Determine the shading direction. Since x is greater than 4, the ray is shaded to the right. Final answer: Open circle at 4, shaded to the right.

7. Maria is organizing a charity fundraiser and needs to rent tables. Each rectangular table can seat 6 people along its length and 4 people along its width. The venue has space for tables with a total perimeter of at most 120 feet. If each table must have a length that is at least 2 feet more than twice its width, write an inequality to represent all possible widths for the tables, then determine the maximum possible width in feet. Answer: 8 Solution: Let w represent the width of each table in feet. The length is at least 2 feet more than twice the width, so length ≥ 2w + 2. The perimeter of one table is 2(length + width).
   Full step-by-step solution

   Step 1: Let w represent the width of each table in feet. Step 2: The length is at least 2 feet more than twice the width, so length ≥ 2w + 2. Step 3: The perimeter of one table is 2(length + width). Step 4: For one table: 2((2w + 2) + w) = 2(3w + 2) = 6w + 4. Step 5: Since the venue has space for tables with total perimeter ≤ 120 feet, and we're finding maximum width for one table, we use the minimum length to maximize width. Step 6: So 6w + 4 ≤ 120 Step 7: Subtract 4 from both sides: 6w ≤ 116 Step 8: Divide both sides by 6: w ≤ 19.33 Step 9: However, we must also consider that length ≥ 2w + 2, and with perimeter constraint, the maximum width occurs when we use the minimum length. Step 10: The inequality 6w + 4 ≤ 120 gives us the maximum possible width. Step 11: 6w ≤ 116 → w ≤ 19.33 Step 12: But we need to check if this satisfies the length condition. With w = 8, length would be at least 18, perimeter = 2(18+8) = 52, which is reasonable. Step 13: The maximum width that satisfies both conditions is 8 feet. The answer is 8.