Using the definition of absolute value there will be two equations that
we will need to set up to get rid of the absolute value, because there
are two places on the number line that are 10 units away from zero: 10
and -10.

***Inv. of add. 2 is sub. 2**

***Inv. of mult. by 7 is div. by 7**

***Inv. of add. 2 is sub. 2**

***Inv. of mult. by 7 is div. by 7**

When we plug 8/7 in for *x*, we end up with
the absolute value of 10 which is 10. When we plug -12/7 in for *x*,
we end up with the absolute value of -10 which is also 10.

If you would have only come up with an answer of *x* = 8/7, you would not have gotten all solutions to this problem.

Again, it is important to note that we are using the definition of absolute
value to set the two equations up. Once you apply the definition
and set it up without the absolute value, you just solve the linear equation
as shown in **Tutorial 14: Linear
Equations in One Variable**.

**There are two solutions to this absolute value equation: 8/7
and -12/7.**

Be careful on this one. It is very tempting to set this up the
same way we did problem 1 above, with two solutions. However, note
that the absolute value is set equal to a negative number. There
is no value of *x* that we can plug in that will
be a solution because when we take the absolute value of the left side
it will always be positive or zero, NEVER negative.

**Answer: No solution.**

Using the definition of absolute value, there will be two equations
that we will need to set up to get rid of the absolute value because there
are two places on the number line that are 8 units away from zero: 8 and
-8.

***Quad. eq. in standard form**

***Factor
the trinomial**

***Use Zero-Product
Principle**

***Set 1st factor = 0 and solve**

***Set 2nd factor = 0 and solve**

***Quad. eq. in standard form**

***Factor
the trinomial**

***Use Zero-Product
Principle**

***Set 1st factor = 0 and solve**

***Set 2nd factor = 0 and solve**

When we plug 6 and -4 in for *x*, we end
up with the absolute value of 8 which is 8. When we plug 4 and -2
in for *x*, we end up with the absolute value
of -8 which is also 8.

If you would have only come up with an answer of *x* = 6 and -4, you would not have gotten all solutions to this problem.

Again, it is important to note that we are using the definition of absolute
value to set the two equations up. Once you apply the definition
and set it up without the absolute value, you just solve the quadratic equation
as shown in **Tutorial 17: Quadratic
Equations**.

**There are four solutions to this absolute value equation: 6, -4,
4, and -2.**

Last revised on Dec. 16, 2009 by Kim Seward.

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