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Graph Inequalities

Grade 7 · Algebra · Worksheet 1

  1. 2(3x - 7) + 5 ≥ 4(x + 2) - 3 Answer: ______________
  2. Emma is organizing a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $75 plus $8 per shirt. Emma's budget for the t-shirts is at most $500. Write an inequality to represent this situation, then determine the maximum number of t-shirts she can order without exceeding her budget. Answer: ______________
  3. Maya is organizing a school fundraiser and needs to order custom t-shirts. The printing company charges a $50 setup fee plus $8 per shirt. Maya's budget for t-shirts is $500. Write an inequality to represent the maximum number of shirts she can order, then solve for the actual maximum number of whole shirts she can purchase. Answer: ______________
  4. A school is organizing a field trip to the science museum. The museum charges a flat fee of $200 for the group plus $12 per student. The school has budgeted at most $800 for this trip. Write an inequality to represent the situation, where s represents the number of students who can attend, then determine the maximum number of students that can go on the trip. Answer: ______________
  5. Hana is graphing the solution to the inequality 3x - 14 > 22 on a number line. What number should she use as the endpoint for her graph, and should she use an open or closed circle? Answer: ______________
  6. 2(3x - 7) ≤ 4x + 10 Answer: ______________
  7. Mere is graphing the solution to the inequality 4x - 6 ≤ 18 on a number line. Solve the inequality and describe the graph, including whether the circle is open or closed and the direction of the arrow. Answer: ______________
  8. Matiu is graphing the solution to the inequality 4x - 7 > 29 on a number line. First, solve the inequality. Then, describe what the graph should look like: Is the circle open or closed? Which direction does the arrow point? Answer: ______________
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Answer Key & Explanations

Graph Inequalities · Grade 7 · Worksheet 1

  1. 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

  2. Emma is organizing a school fundraiser and needs to order custom t-shirts. The printing company charges a flat setup fee of $75 plus $8 per shirt. Emma's budget for the t-shirts is at most $500. Write an inequality to represent this situation, then determine the maximum number of t-shirts she can order without exceeding her budget. Answer: 53 Solution: Let x represent the number of t-shirts. The total cost is the setup fee plus cost per shirt: 75 + 8x Set up the inequality with the budget limit: 75 + 8x ≤ 500 Subtract 75 from both sides: 8x ≤ 425 Divide both sides by 8: x ≤ 53.125 Since we can't order a fraction of a shirt, the maximum number…
    Full step-by-step solution

    Step 1: Let x represent the number of t-shirts. Step 2: The total cost is the setup fee plus cost per shirt: 75 + 8x Step 3: Set up the inequality with the budget limit: 75 + 8x ≤ 500 Step 4: Subtract 75 from both sides: 8x ≤ 425 Step 5: Divide both sides by 8: x ≤ 53.125 Step 6: Since we can't order a fraction of a shirt, the maximum number is 53. The answer is 53.

  3. Maya is organizing a school fundraiser and needs to order custom t-shirts. The printing company charges a $50 setup fee plus $8 per shirt. Maya's budget for t-shirts is $500. Write an inequality to represent the maximum number of shirts she can order, then solve for the actual maximum number of whole shirts she can purchase. Answer: 56 Solution: Let x represent the number of shirts Maya can order. The total cost is the setup fee plus cost per shirt: 50 + 8x Since her budget is $500, we write the inequality: 50 + 8x ≤ 500 Subtract 50 from both sides: 8x ≤ 450 Divide both sides by 8: x ≤ 56.25 Since Maya can only order whole shirts, the…
    Full step-by-step solution

    Step 1: Let x represent the number of shirts Maya can order. Step 2: The total cost is the setup fee plus cost per shirt: 50 + 8x Step 3: Since her budget is $500, we write the inequality: 50 + 8x ≤ 500 Step 4: Subtract 50 from both sides: 8x ≤ 450 Step 5: Divide both sides by 8: x ≤ 56.25 Step 6: Since Maya can only order whole shirts, the maximum number is 56. The answer is 56 shirts.

  4. A school is organizing a field trip to the science museum. The museum charges a flat fee of $200 for the group plus $12 per student. The school has budgeted at most $800 for this trip. Write an inequality to represent the situation, where s represents the number of students who can attend, then determine the maximum number of students that can go on the trip. Answer: 50 Solution: Identify the fixed cost: $200 Identify the variable cost per student: $12 Write the total cost expression: 200 + 12s Set up the inequality for 'at most $800': 200 + 12s ≤ 800 Subtract 200 from both sides: 12s ≤ 600 Divide both sides by 12: s ≤ 50 The maximum number of students is 50.
    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 total cost expression: 200 + 12s Step 4: Set up the inequality for 'at most $800': 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 is 50.

  5. Hana is graphing the solution to the inequality 3x - 14 > 22 on a number line. What number should she use as the endpoint for her graph, and should she use an open or closed circle? Answer: 12, open circle Solution: Solve the inequality 3x - 14 > 22. Add 14 to both sides: 3x - 14 + 14 > 22 + 14 This gives: 3x > 36 Divide both sides by 3: 3x / 3 > 36 / 3 This gives: x > 12 Determine the endpoint and circle type.
    Full step-by-step solution

    Step 1: Solve the inequality 3x - 14 > 22. Add 14 to both sides: 3x - 14 + 14 > 22 + 14 This gives: 3x > 36 Divide both sides by 3: 3x / 3 > 36 / 3 This gives: x > 12 Step 2: Determine the endpoint and circle type. The solution is x > 12, meaning 12 is the boundary number. Since the inequality is strictly greater than (>) and does not include equality, 12 is not part of the solution. Therefore, Hana should use an open circle at 12. Step 3: Graph the solution. On a number line, draw an open circle at 12 and shade the arrow to the right to show all numbers greater than 12. Final answer: 12, open circle

  6. 2(3x - 7) ≤ 4x + 10 Answer: x ≤ 12 Solution: Distribute the 2: 2(3x - 7) = 6x - 14 Rewrite the inequality: 6x - 14 ≤ 4x + 10 Subtract 4x from both sides: 6x - 14 - 4x ≤ 4x + 10 - 4x → 2x - 14 ≤ 10 Add 14 to both sides: 2x - 14 + 14 ≤ 10 + 14 → 2x ≤ 24 Divide both sides by 2: 2x/2 ≤ 24/2 → x ≤ 12 The solution is x ≤ 12.
    Full step-by-step solution

    Step 1: Distribute the 2: 2(3x - 7) = 6x - 14 Step 2: Rewrite the inequality: 6x - 14 ≤ 4x + 10 Step 3: Subtract 4x from both sides: 6x - 14 - 4x ≤ 4x + 10 - 4x → 2x - 14 ≤ 10 Step 4: Add 14 to both sides: 2x - 14 + 14 ≤ 10 + 14 → 2x ≤ 24 Step 5: Divide both sides by 2: 2x/2 ≤ 24/2 → x ≤ 12 The solution is x ≤ 12.

  7. Mere is graphing the solution to the inequality 4x - 6 ≤ 18 on a number line. Solve the inequality and describe the graph, including whether the circle is open or closed and the direction of the arrow. Answer: x ≤ 6, closed circle at 6, arrow to the left Solution: Write the inequality: 4x - 6 ≤ 18 Add 6 to both sides to isolate the term with x: 4x - 6 + 6 ≤ 18 + 6, which simplifies to 4x ≤ 24 Divide both sides by 4 (a positive number, so the inequality sign stays the same): 4x / 4 ≤ 24 / 4, which gives x ≤ 6 Interpret the solution: x ≤ 6 means all numbers…
    Full step-by-step solution

    Step 1: Write the inequality: 4x - 6 ≤ 18 Step 2: Add 6 to both sides to isolate the term with x: 4x - 6 + 6 ≤ 18 + 6, which simplifies to 4x ≤ 24 Step 3: Divide both sides by 4 (a positive number, so the inequality sign stays the same): 4x / 4 ≤ 24 / 4, which gives x ≤ 6 Step 4: Interpret the solution: x ≤ 6 means all numbers less than or equal to 6. On a number line, we place a closed circle at 6 (because 6 is included in the solution) and draw an arrow pointing to the left (to show all numbers less than 6). The answer is x ≤ 6, closed circle at 6, arrow to the left.

  8. Matiu is graphing the solution to the inequality 4x - 7 > 29 on a number line. First, solve the inequality. Then, describe what the graph should look like: Is the circle open or closed? Which direction does the arrow point? Answer: x > 9; open circle at 9, arrow pointing to the right Solution: Start with the inequality 4x - 7 > 29. Add 7 to both sides to isolate the term with x: 4x - 7 + 7 > 29 + 7, which simplifies to 4x > 36. Divide both sides by 4 to solve for x: 4x / 4 > 36 / 4, which gives x > 9.
    Full step-by-step solution

    Step 1: Start with the inequality 4x - 7 > 29. Step 2: Add 7 to both sides to isolate the term with x: 4x - 7 + 7 > 29 + 7, which simplifies to 4x > 36. Step 3: Divide both sides by 4 to solve for x: 4x / 4 > 36 / 4, which gives x > 9. Step 4: To graph x > 9 on a number line, place an open circle at 9 because the inequality is strict (greater than, not greater than or equal to). Then, draw an arrow pointing to the right, indicating all numbers greater than 9. Final answer: The graph has an open circle at 9 and an arrow pointing to the right.