Solve the inequality by adding 3 to both sides to get x < 12.  Since it is absolute value, x – 3 > –9 must also be solved by adding 3 to both sides so: x > –6 so combined.

3w/2 will become greater than 1 as soon as w is greater than two thirds. It will likewise become less than –1 as soon as w is less than negative two thirds. All the other options always return values between –1 and 1.

Absolute value problems always have two sides: one positive and one negative.

First, take the problem as is and drop the absolute value signs for the positive side: z – 3 ≥ 5. When the original inequality is multiplied by –1 we get z – 3 ≤ –5.

Solve each inequality separately to get z ≤ –2 or z ≥ 8 (the inequality sign flips when multiplying or dividing by a negative number).

We can verify the solution by substituting in 0 for z to see if we get a true or false statement. Since –3 ≥ 5 is always false we know we want the two outside inequalities, rather than their intersection.

First, solve the inequality \(\displaystyle 5x-9 \geq6\):

\(\displaystyle 5x-9 \geq6\)

\(\displaystyle 5x\geq15\)

\(\displaystyle x\geq3\)

Since we are dealing with absolute value, \(\displaystyle 5x-9\leq-6\) must also be true; therefore:

\(\displaystyle 5x-9\leq-6\)

\(\displaystyle 5x\leq3\)

\(\displaystyle x\leq \frac{3}{5}\)

To solve \(\displaystyle 3x+2\le 49\), isolate \(\displaystyle x\).

\(\displaystyle 3x\le 47\)

Divide by three on both sides.

\(\displaystyle x\le \frac{47}{3}\)

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation. 

\(\displaystyle -5x+17>62\) Subtract \(\displaystyle 17\) on both sides.

\(\displaystyle -5x>45\) Divide \(\displaystyle -5\) on both sides. Remember to flip the sign.

\(\displaystyle x< -9\)

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation. 

\(\displaystyle 4x+7< 5x+13\) Subtract \(\displaystyle 4x,13\) on both sides.

\(\displaystyle x< -6\)

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation. 

\(\displaystyle \left | x+5\right |< 8\) We need to set-up two equations since its absolute value.

\(\displaystyle x+5< 8\) Subtract \(\displaystyle 5\) on both sides. \(\displaystyle x< 3\)

\(\displaystyle -(x+5)< 8\) Divide \(\displaystyle -1\) on both sides which flips the sign.

\(\displaystyle x+5>-8\) Subtract \(\displaystyle 5\) on both sides. \(\displaystyle x>-13\)

Since we have the \(\displaystyle x\)'s being either greater than or less than the values, we can combine them to get \(\displaystyle -13< x< 3\).

We want to isolate the variable on one side and numbers on another side. Treat like a normal equation. 

\(\displaystyle \left | x+7\right |\leq3x+13\) We need to set-up two equations since it's absolute value.

\(\displaystyle x+7\leq 3x+13\) Subtract \(\displaystyle x, 7\) on both sides.

\(\displaystyle -6\leq 2x\) Divide \(\displaystyle 2\) on both sides.

\(\displaystyle -3\leq x\)

\(\displaystyle -(x+7)\leq 3x+13\) Distribute the negative sign to each term in the parenthesis.

\(\displaystyle -x-7\leq3x+13\) Add \(\displaystyle x\) and subtract \(\displaystyle 13\) on both sides.

\(\displaystyle -20\leq4x\) Divide \(\displaystyle 4\) on both sides.

\(\displaystyle -5\leq x\) We must check each answer. Let's try \(\displaystyle 0\).

\(\displaystyle \left | 0+7\right |< 3(0)+13\)    \(\displaystyle 7< 13\) This is true therefore \(\displaystyle -3\leq x\) is a correct answer. Let's next try \(\displaystyle -4\).

\(\displaystyle \left | -4+7\right |< 3(-4)+13\)  \(\displaystyle 3< 1\) This is not true therefore \(\displaystyle -5\leq x\) is not correct. 

Final answer is just \(\displaystyle -3\leq x\).

To solve this problem, add the two equations together:

\(\displaystyle x+1< 4\)

\(\displaystyle y-2< -1\)

\(\displaystyle x+1+y-2< 4-1\)

\(\displaystyle x+y-1< 3\)

\(\displaystyle x+y< 4\)

The only answer choice that satisfies this equation is 0, because 0 is less than 4.