We simplify this inequality similarly to how we would simplify an equation

Thus 

In order to find the solution set, we solve  as we would an equation:

Therefore, the solution set is any value of .

The inequality that is presented in the problem is:

Start by moving your variables to one side of the inequality and all other numbers to the other side:

Divide both sides of the equation by . Remember to flip the direction of the inequality's sign since you are dividing by a negative number!

Reduce:

The only answer choice with a value greater than  is .

To get the lowest value, you need the lowest numerator and the highest denominator.  That would be  or reduced to be .  For the highest value, you need the highest numerator and the lowest denominator.  That would be  or .

The inequality  can be rewritten as the three-part inequality

Isolate the  in the middle expression by performing the same operations in all three expressions. Subtract 32 from each expression:

Divide each expression by , switching the direction of the inequality symbols:

This can be rewritten in interval notation as .

In an absolute value inequality, the absolute value expression must be isolated first, as follows:

Adding 12 to both sides:

Multiplying both sides by , and switching the inequality symbol due to multiplication by a negative number:

We do not need to go further. An absolute value expression must always be greater than or equal to 0; it is impossible for the expression  to be less than any negative number. The inequality has no solution.

In general, when we take the absolute value of a quantity, we can represent it as either itself, or as its additive inverse.

In other words, |x| = x (if x > 0) or |x| = –x (if x < 0).

Therefore, we can represent |4 – 2x| as either 4 – 2x or as –(4 – 2x). We must consider both of these cases and solve for x. Let’s us first consider the case that |4 – 2x| = 4 – 2x.

4 – 2x < 5

We can add 2x to both sides

4 < 2x + 5

Then subtract 5 from both sides.

–1 < 2x

Divide by 2.

–1/2 < x

x > –0.5

Now, we consider the case that |4 – 2x | = –(4 – 2x).

–(4 – 2x) < 5

Multiply both sides by negative one. Remember, whenever we multiply or divide an inequality by a negative number, we must flip the inequality sign.

4 – 2x > –5

Add 2x to both sides.

4 > 2x – 5

Add 5 to both sides.

9 > 2x

Divide by 2,

9/2 > x

4.5 > x

This means that the values of x that satisfy the original quality must be greater than -0.5 AND less than 4.5.

The question asks us to find the sum of the integer values of x that satisfy the inequality. The only integers between -0.5 and 4.5 are the following: 0, 1, 2, 3, and 4.

The sum of 0, 1, 2, 3, and 4 is 10.

The answer is 10.

There are 3 solutions: 0, 7, 7.

The correct answer is 2 distinct solutions, however, because 7 occurs twice.  So the two distinct solutions are 0 and 7.

No common terms cancel out, and this isn't a difference of squares. 

Let's move the 48 to the other side: x² = 48

Now take the square root of both sides: x = √48 or x = –√48. Don't forget the second (negative) solution! 

Now √48 = √(3*16) = √(3*4²) = 4√3, so the answer is x = 4√3 or x = –4√3.

If , then either

or .  

These two equations yield  and  as answers.