Distribute the four through the binomial of the right side.

\(\displaystyle -8x-7>4(-3x)+4(2)\)

\(\displaystyle -8x-7>-12x+8\)

Add \(\displaystyle 12x\) and \(\displaystyle 7\) on both sides.

\(\displaystyle -8x-7+(12x)+[7]>-12x +(12x)+8+[7]\)

\(\displaystyle 4x>15\)

Divide by four on both sides.

\(\displaystyle \frac{4x}{4}>\frac{15}{4}\)

The answer is:  \(\displaystyle x>\frac{15}{4}\)

Solve.

\(\displaystyle -x+4>2x+2\) 

Step 1: Subtract \(\displaystyle 4\) from both sides of the inequality.

\(\displaystyle -x>2x-2\)

Step 2: Subtract \(\displaystyle 2x\) from both sides of the inequality to isolate the term with the \(\displaystyle x\) variable.

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

Step 3: Multiply both sides of the inequality by -1 and reverse the inequality sign.

This is to make the inequality have only positive numbers, and this will help solve the inequality. Because we are multiplying by a negative number, we must reverse the inequality sign. The only times we reverse the inequality sign are when we are multiplying or dividing by a negative number. In other instances, we would leave the sign the same.

\(\displaystyle 3x< 2\)

Step 4: Divide both sides of the inequality by \(\displaystyle 3\).

\(\displaystyle x< \frac{2}{3}\)

Solution:  \(\displaystyle x< \frac{2}{3}\)

\(\displaystyle -2\leq \frac{1-3x}{4}\leq 4\)

Inequalities can be algebraically rearranged using operations that are mostly identical to algebraic equations, although one notable exception is multiplication or division by -1. This reverses the inequality signs. 

Multiply out by \(\displaystyle 4.\)

\(\displaystyle -8\leq \1-3x\leq 16\)

Subtract \(\displaystyle 1\) from all sides, 

\(\displaystyle -9\leq -3x\leq 15\)

 Divide throughout by \(\displaystyle -3\) and remember to reverse the inequality signs. 

\(\displaystyle 3\geq x\geq -5\)

It feels more natural to write  the final result as: 

\(\displaystyle -5\leq x\leq3\)

Remember: Use inverse operations to undo the operations in the inequality (for example use a subtraction to undo an addition) until you are left with the variable. Make sure to do the same operations to both sides of the inequality. 

Important Note: When multiplying or dividing by a negative number, always flip the sign of an inequality.

Solution:

\(\displaystyle 17-8\times(5-2m)>4m+7+6\times(3m-3)\)

Expand all factors

\(\displaystyle 17-40+16m>4m+7+18m-18\)

Simplify 

\(\displaystyle -23+16m>22m-11\)

Add 23

\(\displaystyle 16m>22m+12\)

Subtract 22m

\(\displaystyle -6m>12\)

Divide by -6 (We flip the sign of the inequality)

\(\displaystyle m< \frac{12}{-6}\)

Simplify

\(\displaystyle m< -2\)

Start by subtracting 1 and divinding by 4 on both sides of the equality

\(\displaystyle 4\leq4x\leq12\)

\(\displaystyle 1\leq x\leq3\)

Written in interval notation:

\(\displaystyle \left[ 1,3\right]\)