\(\displaystyle -12 > z-4\)

Add 4 to both sides of the inequality.

\(\displaystyle -12 + 4> z-4 + 4\)

\(\displaystyle -8 > z\)

Because the variable \(\displaystyle z\) is to the right of the inequality, it needs to be placed to the left of the inequality. When this happens, the inequality sign needs to be switched to its opposite.

\(\displaystyle z < -8\)

\(\displaystyle 3x + 6 \geq -2x + 8\)

Subtract 6 from both sides of the inequality.

\(\displaystyle 3x + 6 - 6\geq -2x + 8 - 6\)

\(\displaystyle 3x\geq -2x + 2\)

Add \(\displaystyle 2x\) to both sides of the inequality.

\(\displaystyle 3x + 2x\geq -2x + 2x + 2\)

\(\displaystyle 5x \geq 2\) 

Divide both sides by \(\displaystyle 5\).

\(\displaystyle \frac{5x}{5} \geq \frac{2}{5}\)

\(\displaystyle x\geq \frac{2}{5}\)

\(\displaystyle n - 6\leq 7\)

Would be the inequality that represents that statement.

Add \(\displaystyle 6\) to both sides of that inequality.

\(\displaystyle n - 6 + 6 \leq 13\)

\(\displaystyle n\leq 13\)

\(\displaystyle x^{2} - 64< 0\)

\(\displaystyle x^{2} < 64\)

\(\displaystyle -8 < x < 8\)

\(\displaystyle 3 (x-3) > 4(x+2)\)

Begin by simplifying the inequality by using the distributive property.

\(\displaystyle 3x - 9 > 4x +8\)

Add \(\displaystyle 9\) to both sides.

\(\displaystyle 3x-9 + 9 > 4x + 8 + 9\)

\(\displaystyle 3x> 4x + 17\)

Subtract \(\displaystyle 4x\) from both sides.

\(\displaystyle 3x-4x > 4x-4x+17\)

\(\displaystyle -x > 17\)

Because there is a negative variable, multiply both sides by \(\displaystyle -1\) and switch the inequality sign to its opposite.

\(\displaystyle x< -17\)

\(\displaystyle b - 6 > 13\)

Add 6 to both sides of the inequality.

\(\displaystyle b-6 + 6> 13 + 6\)

\(\displaystyle b> 19\)

\(\displaystyle -3x-12> 15\)

To isolate the variable, add \(\displaystyle 12\) to both sides of the inequality.

\(\displaystyle -3x-12+12 > 15 + 12\)

\(\displaystyle -3x >27\)

Divide both sides by \(\displaystyle -3.\) Because you are dividing by a negative integer, you have to flip or switch the inequality sign to its opposite.

\(\displaystyle x < -9\)

\(\displaystyle 12+n\leq 4\)

To isolate the variable , subtract \(\displaystyle 12\) from both sides of the inequality.

\(\displaystyle (12-12) + n \leq4-12\)

\(\displaystyle n\leq -8\)

\(\displaystyle x + 0.8\leq -0.5\)

To isolate the variable, subtract \(\displaystyle 0.8\) from both sides of the inequality.

\(\displaystyle x + 0.8-0.8 \leq -0.5-0.8\)

\(\displaystyle x\leq -1.3\)

\(\displaystyle 8x-10 > 4x-6\)

Subtract \(\displaystyle 4x\) from both sides of the equation

\(\displaystyle 8x-4x-10 > 4x-4x-6\)

\(\displaystyle 4x -10> -6\)

Add \(\displaystyle 10\) to both sides of the equation.

\(\displaystyle 4x-10 + 10 = -6 + 10\)

\(\displaystyle 4x > 4\)

Divide both sides by 4.

\(\displaystyle \frac{4x}{4} >\frac{4}{4}\)

\(\displaystyle x>1\)