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

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

Recall that the absolute value sign will convert any value to a positive sign. There will be no occurrences of that will evaluate into a negative one as a final solution.

There are no solutions for this equation.

The answer is:

Below is the graph of :

The given graph is the graph of reflected in the -axis, then translated up 6 units. This graph is

, where .

The function graphed is therefore

Let

The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates . In terms of ,

,

or, alternatively written,

The graph of is the same as that of , after it shifts 10 units left ( ), it flips vertically (negative symbol), and it shifts up 10 units (the second ). The flip does not affect the position of the vertex, but the shifts do; the vertex of the graph of is at .

Below is the graph of :

The given graph is the graph of reflected in the -axis, then translated up 6 units. This graph is

, where .

The function graphed is therefore

Let

The graph of this basic absolute value function is a "V"-shaped graph with a vertex at the origin, or the point with coordinates . In terms of ,

,

or, alternatively written,

The graph of is the same as that of , after it shifts 10 units left ( ), it flips vertically (negative symbol), and it shifts up 10 units (the second ). The flip does not affect the position of the vertex, but the shifts do; the vertex of the graph of is at .

Recall that the absolute value sign will convert any value to a positive sign. There will be no occurrences of that will evaluate into a negative one as a final solution.

There are no solutions for this equation.

The answer is:

Separate the absolute value and solve both the positive and negative components of the absolute value.

Solve the first equation. Add on both sides.

Add two on both sides.

Divide by five on both sides.

One of the solutions is after substitution is valid.

Evaluate the second equation. Distribute the negative sign in the binomial.

Subtract on both sides.

Add two on both sides.

If we substitute this value back to the original equation, the equation becomes invalid.

The answer is:

Separate the absolute value and solve both the positive and negative components of the absolute value.

Solve the first equation. Add on both sides.

Add two on both sides.

Divide by five on both sides.

One of the solutions is after substitution is valid.

Evaluate the second equation. Distribute the negative sign in the binomial.

Subtract on both sides.

Add two on both sides.

If we substitute this value back to the original equation, the equation becomes invalid.

The answer is: