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.  

You can solve this problem using the guess and check method by substituting the first number in the ordered pair for "x" and the second number for "y". Therfore the correct answer is  

–4 = 0 – 4

0 = 4 – 4

12 = 16 – 4

60 = 64 – 4

First, take the square root of both sides:

Therefore,    or   

Add 8 to both sides of the equation; therefore,    or   

This equation can be solved using the guess and check method.

Therefore, the ordered pair (3, 16) is the correct answer.

First multiply each decimal number in each term by 100 to remove the decimals (to get a whole number you have to multiply 0.03 by 100 to get 3).  You need to do this for terms on both sides of the equal sign.

 The second method would be to look for the number of digits to the right of the decimal point (e.g., 0.03 has two digits).  So in this method shift the decimal point to the right two places.

Now the equation looks as follows:

Now solve for  and  will be equal to .