Question

1. $0 < -1 + \frac{n}{12}$
2. $-10 \leq \frac{a}{8} - 9$
3. $-3 > -4 + \frac{k}{9}$
4. $-7 \geq \frac{-4 + p}{2}$
5. $\frac{7 + x}{-2} \geq 2$
6. $3 + 7m \leq -18$
7. $-4n - 1 < -37$
8. $2(-7 + x) > -2$

Explanation:

21) Step1: Isolate the fraction

Add 1 to both sides.
$0 + 1 < -1 + \frac{n}{12} + 1$
$1 < \frac{n}{12}$

21) Step2: Solve for $n$

Multiply by 12 on both sides.
$1 \times 12 < \frac{n}{12} \times 12$
$12 < n$ or $n > 12$

22) Step1: Isolate the fraction

Add 9 to both sides.
$-10 + 9 \leq \frac{a}{8} - 9 + 9$
$-1 \leq \frac{a}{8}$

22) Step2: Solve for $a$

Multiply by 8 on both sides.
$-1 \times 8 \leq \frac{a}{8} \times 8$
$-8 \leq a$ or $a \geq -8$

23) Step1: Isolate the fraction

Add 4 to both sides.
$-3 + 4 > -4 + \frac{k}{9} + 4$
$1 > \frac{k}{9}$

23) Step2: Solve for $k$

Multiply by 9 on both sides.
$1 \times 9 > \frac{k}{9} \times 9$
$9 > k$ or $k < 9$

24) Step1: Eliminate denominator

Multiply by 2 on both sides.
$-7 \times 2 \geq \frac{-4 + p}{2} \times 2$
$-14 \geq -4 + p$

24) Step2: Solve for $p$

Add 4 to both sides.
$-14 + 4 \geq -4 + p + 4$
$-10 \geq p$ or $p \leq -10$

25) Step1: Eliminate denominator

Multiply by -2 (reverse inequality).
$\frac{7+x}{-2} \times (-2) \leq 2 \times (-2)$
$7 + x \leq -4$

25) Step2: Solve for $x$

Subtract 7 from both sides.
$7 + x - 7 \leq -4 - 7$
$x \leq -11$

26) Step1: Isolate the term with $m$

Subtract 3 from both sides.
$3 + 7m - 3 \leq -18 - 3$
$7m \leq -21$

26) Step2: Solve for $m$

Divide by 7 on both sides.
$\frac{7m}{7} \leq \frac{-21}{7}$
$m \leq -3$

27) Step1: Isolate the term with $n$

Add 1 to both sides.
$-4n - 1 + 1 < -37 + 1$
$-4n < -36$

27) Step2: Solve for $n$

Divide by -4 (reverse inequality).
$\frac{-4n}{-4} > \frac{-36}{-4}$
$n > 9$

28) Step1: Simplify left side

Distribute the 2.
$2(-7) + 2x > -2$
$-14 + 2x > -2$

28) Step2: Isolate term with $x$

Add 14 to both sides.
$-14 + 2x + 14 > -2 + 14$
$2x > 12$

28) Step3: Solve for $x$

Divide by 2 on both sides.
$\frac{2x}{2} > \frac{12}{2}$
$x > 6$

Answer:

1. $n > 12$ (Graph: Open circle at 12, arrow right)
2. $a \geq -8$ (Graph: Closed circle at -8, arrow right)
3. $k < 9$ (Graph: Open circle at 9, arrow left)
4. $p \leq -10$ (Graph: Closed circle at -10, arrow left)
5. $x \leq -11$ (Graph: Closed circle at -11, arrow left)
6. $m \leq -3$ (Graph: Closed circle at -3, arrow left)
7. $n > 9$ (Graph: Open circle at 9, arrow right)
8. $x > 6$ (Graph: Open circle at 6, arrow right)