Graph Inequalities

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Answer Key & Explanations

Graph Inequalities · Grade 7 · Worksheet 2

1. 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.

2. Emma is organizing a school fundraiser and needs to order custom t-shirts. The printing company charges a flat fee of $150 for setup 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 Emma 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 flat fee plus the cost per shirt: 150 + 8x The budget limit means the total cost must be less than or equal to 1000: 150 + 8x ≤ 1000 Subtract 150 from both sides: 8x ≤ 850 Divide both sides by 8: x ≤ 106.25 Since Emma…
   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 flat fee plus the cost per shirt: 150 + 8x Step 3: The budget limit means the total cost must be less than or equal to 1000: 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 cannot order a fraction of a shirt, the maximum number is 106. The answer is 106.

3. 2(3x - 7) + 5 ≥ 4(x + 2) - 3 Answer: x ≥ 8 Solution: Distribute the 2 on the left side: 2 × 3x = 6x and 2 × (-7) = -14, so we get 6x - 14 + 5 ≥ 4(x + 2) - 3 Simplify the left side: -14 + 5 = -9, so we have 6x - 9 ≥ 4(x + 2) - 3 Distribute the 4 on the right side: 4 × x = 4x and 4 × 2 = 8, so we get 6x - 9 ≥ 4x + 8 - 3 Simplify the right side: 8 -…
   Full step-by-step solution

   Step 1: Distribute the 2 on the left side: 2 × 3x = 6x and 2 × (-7) = -14, so we get 6x - 14 + 5 ≥ 4(x + 2) - 3 Step 2: Simplify the left side: -14 + 5 = -9, so we have 6x - 9 ≥ 4(x + 2) - 3 Step 3: Distribute the 4 on the right side: 4 × x = 4x and 4 × 2 = 8, so we get 6x - 9 ≥ 4x + 8 - 3 Step 4: Simplify the right side: 8 - 3 = 5, so we have 6x - 9 ≥ 4x + 5 Step 5: Subtract 4x from both sides: 6x - 4x - 9 ≥ 5, which gives 2x - 9 ≥ 5 Step 6: Add 9 to both sides: 2x ≥ 5 + 9, which gives 2x ≥ 14 Step 7: Divide both sides by 2: x ≥ 7 Step 8: The solution is x ≥ 7

4. Graph 5(2x - 10) + 15 ≥ 10x - 35 Answer: x ≥ -5 Solution: Distribute the 5: 5(2x - 10) = 10x - 50. The inequality becomes 10x - 50 + 15 ≥ 10x - 35. Combine like terms on the left: -50 + 15 = -35, so 10x - 35 ≥ 10x - 35.
   Full step-by-step solution

   Step 1: Distribute the 5: 5(2x - 10) = 10x - 50. The inequality becomes 10x - 50 + 15 ≥ 10x - 35. Step 2: Combine like terms on the left: -50 + 15 = -35, so 10x - 35 ≥ 10x - 35. Step 3: Subtract 10x from both sides: 10x - 35 - 10x ≥ 10x - 35 - 10x → -35 ≥ -35. Step 4: This is a true statement (-35 is equal to -35), so the inequality holds for all real numbers. However, note that the original inequality simplifies to an identity. Let's re-check: Actually, after subtracting 10x from both sides, we get -35 ≥ -35, which is true. This means the solution is all real numbers. But wait, the problem asks to graph the solution. Since it's all real numbers, the number line is completely shaded. However, the answer format expects a specific inequality. Let's re-solve carefully. Actually, let's re-examine: 10x - 35 ≥ 10x - 35. Subtract 10x from both sides: -35 ≥ -35. This is always true, so the solution is all real numbers. But the problem likely expects a specific x value. Let's check if there's a mistake: The original inequality is 5(2x - 10) + 15 ≥ 10x - 35. Distribute: 10x - 50 + 15 ≥ 10x - 35 → 10x - 35 ≥ 10x - 35. This is an identity, so the solution is all real numbers. However, to match the expected answer format, we can write x ∈ ℝ. But since the instruction says to graph on a number line, the graph would be a line shaded everywhere. For the answer, we can state: All real numbers. But wait, the problem might have been intended to have a different constant. Let's re-check the numbers: 5(2x - 10) + 15 ≥ 10x - 35. If we move terms: 10x - 35 ≥ 10x - 35, subtract 10x: -35 ≥ -35, true. So the solution is all real numbers. The answer is x ∈ ℝ. The graph is a number line with the entire line shaded. Final answer: All real numbers (x ∈ ℝ).

5. Maya is organizing a charity walkathon where participants collect pledges based on distance walked. The event has a $200 setup fee plus $15 per participant for supplies. If the total budget cannot exceed $1,000, write an inequality to represent the maximum number of participants, p, that can register, then solve for p. Answer: 53 Solution: Identify the fixed cost: $200 setup fee Identify the variable cost: $15 per participant Write the total cost expression: 200 + 15p Set up the inequality for 'cannot exceed $1,000': 200 + 15p ≤ 1000 Subtract 200 from both sides: 15p ≤ 800 Divide both sides by 15: p ≤ 800/15 Calculate 800 ÷ 15 =…
   Full step-by-step solution

   Step 1: Identify the fixed cost: $200 setup fee Step 2: Identify the variable cost: $15 per participant Step 3: Write the total cost expression: 200 + 15p Step 4: Set up the inequality for 'cannot exceed $1,000': 200 + 15p ≤ 1000 Step 5: Subtract 200 from both sides: 15p ≤ 800 Step 6: Divide both sides by 15: p ≤ 800/15 Step 7: Calculate 800 ÷ 15 = 53.333... Step 8: Since we can't have a fraction of a person, the maximum whole number is 53 Step 9: Check: 200 + 15(53) = 200 + 795 = 995, which is ≤ 1000 Step 10: For 54 participants: 200 + 15(54) = 200 + 810 = 1010, which exceeds the budget Therefore, the maximum number of participants is 53.

6. Aroha is buying supplies for a school project. Each package of markers costs $3. Aroha has a budget of $31 to spend on markers. What is the maximum number of packages Aroha can buy? Graph the solution on a number line. Answer: 10 Solution: Let p = number of packages. Inequality: 3p ≤ 31. Solve: p ≤ 31 / 3 = 10.
   Full step-by-step solution

   Step 1: Let p = number of packages. Inequality: 3p ≤ 31. Step 2: Solve: p ≤ 31 / 3 = 10. Step 3: Since p must be a whole number, the maximum is 10. Step 4: Graph: closed circle at 10, arrow to the left.

7. 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

8. Liam is graphing the solution to the inequality 5x + 15 ≤ 40 on a number line. Solve the inequality, then describe how the graph should look: include the type of circle (open or closed) and the direction of the arrow. Answer: x ≤ 5; closed circle at 5, arrow pointing left Solution: Since x is less than or equal to 5, the arrow points to the left (towards smaller numbers).
   Full step-by-step solution

   Step 1: Start with the inequality: 5x + 15 ≤ 40 Step 2: Subtract 15 from both sides: 5x + 15 - 15 ≤ 40 - 15, which simplifies to 5x ≤ 25 Step 3: Divide both sides by 5 (positive, so inequality sign stays the same): 5x ÷ 5 ≤ 25 ÷ 5, which gives x ≤ 5 Step 4: On the number line, we place a closed circle at 5 because the inequality includes 'or equal to' (≤). Step 5: Since x is less than or equal to 5, the arrow points to the left (towards smaller numbers). The answer is x ≤ 5; closed circle at 5, arrow pointing left.