Two-Step Inequalities

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Answer Key & Explanations

Two-Step Inequalities · Grade 7 · Worksheet 2

1. 3x + 7 > 22 Answer: x > 5 Solution: 3x + 7 > 22 Subtract 7 from both sides to isolate the term with x. This is because we want to undo the addition of 7. 3x + 7 - 7 > 22 - 7 3x > 15 Divide both sides by 3 to solve for x.
   Full step-by-step solution

   We start with the inequality: 3x + 7 > 22 Step 1: Subtract 7 from both sides to isolate the term with x. This is because we want to undo the addition of 7. 3x + 7 - 7 > 22 - 7 3x > 15 Step 2: Divide both sides by 3 to solve for x. Since 3 is positive, the inequality sign stays the same. 3x / 3 > 15 / 3 x > 5 So the solution is x > 5.

2. Noah draws a number line and shades the solution region for the inequality 9x - 15 ≤ 75. On the number line, he marks a closed circle at the boundary value. What is the boundary value where Noah places the closed circle? Answer: 10 Solution: Start with the inequality 9x - 15 ≤ 75. Add 15 to both sides to isolate the term with x: 9x - 15 + 15 ≤ 75 + 15, which simplifies to 9x ≤ 90.
   Full step-by-step solution

   Step 1: Start with the inequality 9x - 15 ≤ 75. Step 2: Add 15 to both sides to isolate the term with x: 9x - 15 + 15 ≤ 75 + 15, which simplifies to 9x ≤ 90. Step 3: Divide both sides by 9 to solve for x: 9x / 9 ≤ 90 / 9, which simplifies to x ≤ 10. Step 4: The boundary value is 10, and since the inequality is ≤, Noah places a closed circle at 10 on the number line. The answer is 10.

3. Matiu is helping his school's technology club raise money to buy new robotics kits. The club already has $124 saved from previous fundraisers. They plan to sell handmade candles at the school fair for $8 each. The robotics kits cost $300 in total. Write and solve an inequality to find the minimum number of candles, c, Matiu and his club must sell to have enough money to buy the kits. Answer: 22 Solution: Let c represent the number of candles sold. The money from selling candles is 8c dollars. The total money is 124 + 8c.
   Full step-by-step solution

   Step 1: Let c represent the number of candles sold. Step 2: The money from selling candles is 8c dollars. The total money is 124 + 8c. Step 3: The total money must be at least $300, so the inequality is: 124 + 8c >= 300 Step 4: Subtract 124 from both sides: 8c >= 176 Step 5: Divide both sides by 8: c >= 22 Step 6: Since c must be a whole number, the minimum number of candles to sell is 22. The answer is 22.

4. 5x - 13 > 27. Solve and graph. Answer: x > 8 Solution: Add 13 to both sides: 5x - 13 + 13 > 27 + 13 becomes 5x > 40. Divide both sides by 5: 5x/5 > 40/5 becomes x > 8. The solution is x > 8.
   Full step-by-step solution

   Step 1: Add 13 to both sides: 5x - 13 + 13 > 27 + 13 becomes 5x > 40. Step 2: Divide both sides by 5: 5x/5 > 40/5 becomes x > 8. The solution is x > 8. To graph: Draw a number line, place an open circle at 8 (since 8 is not included), and shade the arrow to the right.

5. A rectangular garden is being planned with a length of 15 meters and a width of 8 meters. A stone path of uniform width 'x' meters will be built around the entire garden, inside a larger rectangular fence. The area of the stone path alone must be at least 74 square meters. Write and solve the inequality to find the minimum width 'x' of the path. Answer: x ≥ 2 Solution: This type of problem involves creating an inequality from a geometric scenario. The key is to express the area of a border (like a path or a frame) by finding the difference between the total area and the inner area.
   Full step-by-step solution

   This type of problem involves creating an inequality from a geometric scenario. The key is to express the area of a border (like a path or a frame) by finding the difference between the total area and the inner area. After setting up the equation, you simplify it into a standard quadratic form and solve for the variable, remembering to interpret the solution in the context of the real-world constraint.

6. Mere is drawing a number line to graph the solution to the inequality 7x - 24 > 60. She marks a point on the number line at the boundary value. Should she use an open or closed circle at that point, and in which direction should she shade the number line? Answer: Open circle at 12, shade to the right Solution: Solve the inequality 7x - 24 > 60. Add 24 to both sides: 7x - 24 + 24 > 60 + 24, so 7x > 84. Divide both sides by 7: 7x / 7 > 84 / 7, so x > 12.
   Full step-by-step solution

   Step 1: Solve the inequality 7x - 24 > 60. Add 24 to both sides: 7x - 24 + 24 > 60 + 24, so 7x > 84. Divide both sides by 7: 7x / 7 > 84 / 7, so x > 12. Step 2: Interpret the solution x > 12. Since the inequality is strictly greater than ( > ), the boundary value 12 is not included in the solution set. Therefore, we use an open circle at 12 on the number line. Step 3: Determine the shading direction. The solution is x > 12, which means all numbers greater than 12. On a number line, numbers increase to the right, so we shade to the right of 12. The answer is: open circle at 12, shade to the right.