Question

1. $6 - 4(6n + 7) \geq 122$
2. $-138 \geq -6(6b - 7)$
3. $167 < 6 + 7(2 - 7x)$
4. $5(6 + 3k) + 7 \geq 127$
5. $-8x + 2x - 16 < -5x + 7x$
6. $-1 - 6x - 6 > -11 - 7x$

Explanation:

Problem 7: $6 - 4(6n + 7) \geq 122$

Step1: Expand the parentheses

$6 - 24n - 28 \geq 122$

Step2: Combine constant terms

$-22 - 24n \geq 122$

Step3: Isolate the variable term

$-24n \geq 122 + 22$
$-24n \geq 144$

Step4: Solve for n (reverse inequality)

$n \leq \frac{144}{-24}$
$n \leq -6$

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Problem 8: $-138 \geq -6(6b - 7)$

Step1: Expand the parentheses

$-138 \geq -36b + 42$

Step2: Isolate the variable term

$36b \geq 42 + 138$
$36b \geq 180$

Step3: Solve for b

$b \geq \frac{180}{36}$
$b \geq 5$

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Problem 9: $167 < 6 + 7(2 - 7r)$

Step1: Expand the parentheses

$167 < 6 + 14 - 49r$

Step2: Combine constant terms

$167 < 20 - 49r$

Step3: Isolate the variable term

$-49r > 167 - 20$
$-49r > 147$

Step4: Solve for r (reverse inequality)

$r < \frac{147}{-49}$
$r < -3$

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Problem 10: $5(6 + 3k) + 7 \geq 127$

Step1: Expand the parentheses

$30 + 15k + 7 \geq 127$

Step2: Combine constant terms

$37 + 15k \geq 127$

Step3: Isolate the variable term

$15k \geq 127 - 37$
$15k \geq 90$

Step4: Solve for k

$k \geq \frac{90}{15}$
$k \geq 6$

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Problem 11: $-8x + 2x - 16 < -5x + 7x$

Step1: Combine like terms on both sides

$-6x - 16 < 2x$

Step2: Isolate the variable term

$-6x - 2x < 16$
$-8x < 16$

Step3: Solve for x (reverse inequality)

$x > \frac{16}{-8}$
$x > -2$

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Problem 12: $-1 - 6x - 6 > -11 - 7x$

Step1: Combine constant terms

$-6x - 7 > -11 - 7x$

Step2: Isolate the variable term

$-6x + 7x > -11 + 7$
$x > -4$

Answer:

1. $n \leq -6$
2. $b \geq 5$
3. $r < -3$
4. $k \geq 6$
5. $x > -2$
6. $x > -4$

(For graphing: For each solution, plot a closed dot for $\leq/\geq$ and open dot for $$, then shade the appropriate direction on the number line.)