Start by subtracting 13 from each side. This gives us . Then subtract  from each side. This gives us . Divide both sides by 2 to get . Therefore all values of  where  will satisfy the original inequality.

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.

To solve this problem, add the two equations together:

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

First, solve the inequality :

Since we are dealing with absolute value,  must also be true; therefore:

Given a fractional algebraic equation with variables in the numerator and denominator of one side and the other side equal to zero, we rely on a simple concept.  Zero divided by anything equals zero. That means we can focus in on what values make the numerator (the top part of the fraction) zero, or in other words,

The expression  is a difference of squares that can be factored as 

Solving this for  gives either  or .  That means either of these values will make our numerator equal zero.  We might be tempted to conclude that both are valid answers.  However, our statement earlier that zero divided by anything is zero has one caveat. We can never divide by zero itself.  That means that any values that make our denominator zero must be rejected.  Therefore we must also look at the denominator.

The left side factors as follows

This means that if  is  or , we end up dividing by zero.  That means that  cannot be a valid solution, leaving  as the only valid answer.  Therefore only #3 is correct. 

The absolute value will always be positive or 0, therefore all values of z will create a true statement as long as . Thus all values except for 2 will work.

The denominator of a fraction can never be equal to 0.

Therefore, to find out what x cannot be equal to, we must factor the denominator, and determine what values of x would make it equal to 0. 

Therefore,  and .

You cannot take the square root of a negative number.

Setting up the inequality we get:

Solving for  we get:

Therefore any value less than four will not work, .

Another approach is to plug in each of the possible values.

When plugged into  all of the answers give us a value greater than or equal to 0, except for , which gives us .