### All Algebra II Resources

## Example Questions

### Example Question #71 : Solving Absolute Value Equations

Solve.

**Possible Answers:**

**or **

**and **

**and **

**or **

**or **

**Correct answer:**

**or **

Solve.

Step 1: **Isolate the absolute value by subtracting from both sides of the equation.**

Step 2:** Divide -1 from both sides of the equation in order to get rid of the negative sign in front of the absolute value.**

Step 3: **Because this is an inequality, this equation can be solved in two parts as shown below.**

**Note**: can be written as or

**or **

Step 4: **Solve both parts.**

**Distribute the .**

**Subtract from both sides of the equation.**

**Divide both sides of the equation by .**

**Distribute the .**

**Subtract from both sides of the equation.**

**Divide both sides of the equation by .**

Step 5: **Combine both parts using "or".**

** or **

**Solution: or **

### Example Question #72 : Solving Absolute Value Equations

Solve for values given the equation

**Possible Answers:**

and

and

and

**Correct answer:**

and

Given:

When given an absolute value recognize there are often multiple solutions. The reason why is best exhibited in a simpler example:

Given solve for values of that make this statement true. When you taken an absolute value of something you always end up with the positive number so both and would make this statement true. The solutions can also be written as .

In the case of the more complicated equation for the same reason there are potentially two solutions, which are shown by as an absolute value will always end up creating a positive result.

**To simplify the absolute value we must look at each of these cases:**

** Let's start with the positive case:**

Just like a normal equation with one unknown we will simplify it by isolating by itself. This is first done by subtracting from both sides leaving:

Next is divided from both sides leaving** **, as your final solution.

To check this solution it must be substituted in the original absolute value for and if it's a correct answer you'll end up with a true statement, so

,

so this becomes:

and the absolute value of is so you end up with a true statement. **Therefore **is a **valid solution**

** Next let's solve for the negative case:**

Distribute the negative sign, which is just to make calculations easier and you'll get:

Next can be added to both sides, giving

dividing by leaves:

Checking this solution is done just as you did for the previous solution obtained.

Given ,

substitute in for

multiply gives

so you obtain

adding

and the absolute value of is thereby making this also a valid solution, therefore the two valid solutions are ** and **

### Example Question #73 : Solving Absolute Value Equations

Solve for .

**Possible Answers:**

and

and

and

**Correct answer:**

and

Solve for given

When given an absolute value equation recognize there are often multiple solutions. The reason why is best exhibited in a simpler example:

Given solve for values of that make this statement true. When you taken an absolute value of something you always end up with the positive number so both and would make this statement true. The solutions can also be written as ±.

In the case of the more complicated equation , however before proceeding let's simplify this equation a little as there are two terms that can be added together within the absolute value. These are and .

When added together this gives ,

thereby giving you

For the same reason as shown in the case of there are potentially two solutions to this equation, which are shown by as an absolute value will always end up creating a positive result. **Make sure you apply the parentheses to only the portion of the equation with the absolute value otherwise your answer will be incorrect.**

**To simplify the absolute value we must look at each of these cases:**

** Let's start with the positive case:**

Just like a normal equation with one unknown we will simplify it by isolating by itself. This is first done by combining like terms and getting on one side of the equation.

We can first subtract from which is , this gives you:

Next we can subtract from both sides of the equation.

Dividing both sides by gives you a final answer of this however can be simplified to as both the numerator and denominator are divisible by .

**so **

To check this solution it must be substituted in the original absolute value for and if it's a correct answer you'll end up with a true statement, so

,

Simplify the equation by multiplying by and by

This leaves our equation with

Next add to both sides of the equation:

In order to simplify you must find a common denominator, which is most easily done by multiplying .

This leaves:

Similarly a common denominator is found for and by multiplying by which gives you

this leaves:

simplifying further gives you

which is a true statement, so is a valid solution

**Next let's solve for the negative case:**

Distribute the negative sign, which is just to make calculations easier and you'll get:

Combine like terms:

This can be simplified to as both the numerator and denominator are divisible by , therefore you final answer is

Checking this solution is done just as you did for the previous solution obtained.

Given

substitute in for

Multiplying by gives

Multiplying gives

So the equation simplifies to:

Next can be added to both sides of the equation giving:

Now common denominators must be found for and

The common denominator for is found by multiplying by . This gives you . This can then be simplified through addition and gives .

The common denominator of is found by multiplying and

so

The simplified equation becomes

Through dividing by you get and the absolute value of is , so you get . This is true statement

so is also a valid solution.

### Example Question #5 : Solving Absolute Value Equations

Solve for :

**Possible Answers:**

**Correct answer:**

To solve absolute value equations, we must understand that the absoute value function makes a value positive. So when we are solving these problems, we must consider two scenarios, one where the value is positive and one where the value is negative.

and

This gives us:

and

However, this question has an outside of the absolute value expression, in this case . Thus, any negative value of will make the right side of the equation equal to a negative number, which cannot be true for an absolute value expression. Thus, is an extraneous solution, as cannot equal a negative number.

Our final solution is then

### Example Question #74 : Solving Absolute Value Equations

Which values of provide the full solution set for the inequality:

**Possible Answers:**

**Correct answer:**

Certified Tutor

Certified Tutor