Inequality Solutions

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Inequality Solutions · Grade 6 · Worksheet 3

1. Kaia draws a number line and shades all points to the right of -5, including -5 itself. Which inequality represents the shaded region, and is x = 3 a solution to this inequality? Answer: x ≥ -5; yes Solution: The shading includes -5 (closed circle or filled dot) and all numbers to the right (greater than -5). This is represented by the inequality x ≥ -5.
   Full step-by-step solution

   Step 1: The shading includes -5 (closed circle or filled dot) and all numbers to the right (greater than -5). Step 2: This is represented by the inequality x ≥ -5. Step 3: To check if x = 3 is a solution, substitute 3 into the inequality: 3 ≥ -5. Step 4: Since 3 is greater than -5, the inequality is true. Step 5: Therefore, x = 3 is a solution. The answer is x ≥ -5; yes.

2. Aroha is drawing a number line to represent the inequality x > -5. She marks -5 with an open circle and shades to the right. Then she asks: 'If I test the value x = -3, is it a solution to this inequality? Explain how the number line shows this.' What should Aroha answer? Answer: Yes, x = -3 is a solution. Solution: This means any number greater than -5 is a solution. The symbol > means 'greater than', and -5 is the boundary, which is not included (hence the open circle). Numbers increase as you move to the right.
   Full step-by-step solution

   Step 1: Understand the inequality x > -5. This means any number greater than -5 is a solution. The symbol > means 'greater than', and -5 is the boundary, which is not included (hence the open circle). Step 2: Picture the number line. Numbers increase as you move to the right. So -5 is at a point, and all numbers to its right (like -4, -3, -2, -1, 0, 1, etc.) are greater than -5. Step 3: Locate x = -3 on the number line. Since -3 is to the right of -5, it is greater than -5. Step 4: Check if -3 is in the shaded region. The shading covers all numbers to the right of -5 (but not -5 itself). Since -3 is to the right of -5, it lies in the shaded region. Step 5: Therefore, x = -3 satisfies the inequality x > -5. The answer is: Yes, x = -3 is a solution.

3. If 3x - 7 > 14, solve for x. Answer: x > 7 Solution: 3x - 7 > 14 Add 7 to both sides of the inequality. This is done to isolate the term with x on one side. Adding 7 to both sides gives: 3x - 7 + 7 > 14 + 7 3x > 21 Divide both sides by 3.
   Full step-by-step solution

   We start with the inequality: 3x - 7 > 14 Step 1: Add 7 to both sides of the inequality. This is done to isolate the term with x on one side. Adding 7 to both sides gives: 3x - 7 + 7 > 14 + 7 3x > 21 Step 2: Divide both sides by 3. This will solve for x. Since 3 is positive, the inequality sign remains the same. 3x / 3 > 21 / 3 x > 7 Final answer: x > 7

4. A school is organizing a field trip and needs to rent buses. Each bus can hold 48 students, and there must be at least 3 chaperones per bus. The school has 285 students signed up for the trip and 18 chaperones available. Write an inequality to determine the minimum number of buses, b, needed to accommodate all students while meeting the chaperone requirement. Answer: b ≥ 7 Solution: For transportation scenarios, both passenger capacity and supervision ratios often need to be considered to find a solution that meets all conditions.
   Full step-by-step solution

   When solving real-world problems with multiple constraints, you need to consider all requirements simultaneously. For transportation scenarios, both passenger capacity and supervision ratios often need to be considered to find a solution that meets all conditions.

5. Liam is saving money to buy a new video game that costs $45. He already has $18 saved from his allowance. Each week, he earns $9 from doing chores. Write an inequality to determine how many more weeks, w, Liam needs to save to have at least enough money for the game. Answer: 18 + 9w ≥ 45 Solution: - Cost of video game: $45 - Money already saved: $18 - Money earned per week from chores: $9 Write an expression for total money saved after \( w \) weeks Money saved = money already saved + money earned in \( w \) weeks Money saved = \( 18 + 9 \times w \) That is: \( 18 + 9w \) We want Liam to…
   Full step-by-step solution

   Let's break this down step by step. --- **Step 1: Identify what we know** - Cost of video game: $45 - Money already saved: $18 - Money earned per week from chores: $9 - Number of weeks needed: \( w \) --- **Step 2: Write an expression for total money saved after \( w \) weeks** Money saved = money already saved + money earned in \( w \) weeks Money saved = \( 18 + 9 \times w \) That is: \( 18 + 9w \) --- **Step 3: Set up the inequality** We want Liam to have **at least** enough money for the game. "At least" means greater than or equal to the cost. So: \[ 18 + 9w \geq 45 \] --- **Step 4: Final check** - \( 18 \) is the starting amount - \( 9w \) is the additional money from \( w \) weeks of chores - \( \geq 45 \) ensures he can buy the game (possibly with extra money) --- **Final inequality:** 18 + 9w ≥ 45

6. Is x = 8 a solution to x ≤ 8? Answer: Yes Solution: Write the inequality: x ≤ 8 Substitute x = 8 into the inequality: 8 ≤ 8 The symbol ≤ means 'less than or equal to'. Since 8 is equal to 8, the statement is true.
   Full step-by-step solution

   Step 1: Write the inequality: x ≤ 8 Step 2: Substitute x = 8 into the inequality: 8 ≤ 8 Step 3: The symbol ≤ means 'less than or equal to'. Since 8 is equal to 8, the statement is true. Therefore, x = 8 is a solution to x ≤ 8. Answer: Yes

7. Isabella is packing gift bags for her birthday party. She has a box of chocolates that originally contained 72 pieces. After eating a few, she has at most 57 pieces left. Isabella writes the inequality x ≤ 57 to represent the number of pieces, x, that are left. Which of the following numbers on the number line are possible solutions for x? Test the values 52, 57, 62, and 47. Circle all that are solutions. Answer: 52, 57, 47 Solution: - 52: Is 52 ≤ 57? Yes, because 52 is less than 57. - 57: Is 57 ≤ 57?
   Full step-by-step solution

   Step 1: Understand the inequality: x ≤ 57 means x can be any number that is less than or equal to 57. Step 2: Test each value: - 52: Is 52 ≤ 57? Yes, because 52 is less than 57. - 57: Is 57 ≤ 57? Yes, because 57 is equal to 57. - 62: Is 62 ≤ 57? No, because 62 is greater than 57. - 47: Is 47 ≤ 57? Yes, because 47 is less than 57. Step 3: The solutions are 52, 57, and 47. The answer is 52, 57, 47.

8. If 3x - 8 > 25, what is the smallest integer value of x? Answer: 12 Solution: Start with the inequality: 3x - 8 > 25 Add 8 to both sides to isolate the term with x: 3x - 8 + 8 > 25 + 8 Simplify: 3x > 33 Divide both sides by 3 to solve for x: 3x ÷ 3 > 33 ÷ 3 Simplify: x > 11 Since x must be greater than 11, the smallest integer that satisfies this is 12.
   Full step-by-step solution

   Step 1: Start with the inequality: 3x - 8 > 25 Step 2: Add 8 to both sides to isolate the term with x: 3x - 8 + 8 > 25 + 8 Step 3: Simplify: 3x > 33 Step 4: Divide both sides by 3 to solve for x: 3x ÷ 3 > 33 ÷ 3 Step 5: Simplify: x > 11 Step 6: Since x must be greater than 11, the smallest integer that satisfies this is 12. The smallest integer value of x is 12.