Question

solve each inequality and graph its solution. 17. $9|x - 1| + 6 < 69$ 18. $7|2x - 10| \geq 14$ 19. the temperature at noon in los angeles on a summer day was $88^{\circ}$f. during the day, the temperature varied from this by as much as $7.5^{\circ}$f. write and solve an absolute - value inequality to find the range of possible temperatures for that day.

Explanation:

Step1: Isolate absolute value term

Subtract 6 from both sides:
$9|x-1| < 69 - 6$
$9|x-1| < 63$
Divide by 9:
$|x-1| < 7$

Step2: Rewrite without absolute value

For $|A| < B$, $-B < A < B$:
$-7 < x - 1 < 7$

Step3: Solve for x

Add 1 to all parts:
$-7 + 1 < x < 7 + 1$
$-6 < x < 8$

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Step1: Isolate absolute value term

Divide both sides by 7:
$|2x - 10| \geq \frac{14}{7}$
$|2x - 10| \geq 2$

Step2: Rewrite without absolute value

For $|A| \geq B$, $A \leq -B$ or $A \geq B$:
$2x - 10 \leq -2$ or $2x - 10 \geq 2$

Step3: Solve first inequality

Add 10, then divide by 2:
$2x \leq -2 + 10$
$2x \leq 8$
$x \leq 4$

Step4: Solve second inequality

Add 10, then divide by 2:
$2x \geq 2 + 10$
$2x \geq 12$
$x \geq 6$

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Step1: Define variable & write inequality

Let $x$ = temperature. Variation ≤7.5 from 88:
$|x - 88| \leq 7.5$

Step2: Rewrite without absolute value

$-7.5 \leq x - 88 \leq 7.5$

Step3: Solve for x

Add 88 to all parts:
$88 - 7.5 \leq x \leq 88 + 7.5$
$80.5 \leq x \leq 95.5$

Answer:

1. $-6 < x < 8$
2. $x \leq 4$ or $x \geq 6$
3. The absolute-value inequality is $|x - 88| \leq 7.5$, and the range of possible temperatures is $80.5^\circ\text{F} \leq x \leq 95.5^\circ\text{F}$