Question

1 priya looks at the inequality $12 - x > 5$ and says \i subtract a number from 12 and want a result that is bigger than 5. that means that the solutions should be values of $x$ that are smaller than something.\ do you agree with priya? explain your reasoning and include solutions to the inequality in your explanation. 2 each day, a store opens with 175 shirts out on display. on average, 6 shirts are sold per hour. the store manager likes to keep at least 150 shirts on display at all times. if there are fewer than 150 on display, more shirts are brought out from the storeroom. the manager writes the inequality $-6x + 175 \geq 150$ to describe the situation. a. explain what $x$ represents in the inequality. b. solve the inequality. c. explain what the solution means in the situation.

Explanation:

(Problem 1):

Step1: Isolate the variable term

Subtract 12 from both sides:
$12 - x - 12 > 5 - 12$
$-x > -7$

Step2: Solve for x (reverse inequality)

Multiply both sides by -1:
$x < 7$

Step3: Verify Priya's reasoning

The solution is all x smaller than 7, which matches Priya's statement that solutions are values of x smaller than something.

(Problem 2a):
No calculation needed; interpret the inequality components:
The term $-6x$ represents the total number of shirts sold over time, 6 shirts per hour. So $x$ stands for the number of hours the store has been open.

(Problem 2b):

Step1: Isolate the variable term

Subtract 175 from both sides:
$-6x + 175 - 175 \geq 150 - 175$
$-6x \geq -25$

Step2: Solve for x (reverse inequality)

Divide both sides by -6:
$x \leq \frac{25}{6}$
$x \leq 4\frac{1}{6}$

(Problem 2c):
No calculation needed; connect the solution to the real-world context:
The solution $x \leq 4\frac{1}{6}$ means the store can operate for up to $4\frac{1}{6}$ hours (or 4 hours and 10 minutes) before the number of shirts on display drops below 150, requiring more shirts to be brought out from the storeroom.

Answer:

(Problem 1):
Yes, I agree with Priya. When solving $12 - x > 5$, we find the solution is $x < 7$, meaning all values of x smaller than 7 satisfy the inequality, which aligns with her reasoning.

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