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Absolute Value Equations

To solve the absolute value equation of the form

$$\left| x \right| = k$$

For this to work, we can't allow the constant \(k\) to be a negative number, thus we need the necessary condition

$$k \ge 0$$

Then, we can break it up into two equations

$$x = + k$$

or

$$x = - k$$

Notice that on the left side, we simply drop the absolute value sign but we have to attach a positive sign \(\left( + \right)\) to one equation and a negative sign \(\left( - \right)\) to another. After separating the absolute value equation into two linear equations, we then proceed to solve each.

Example 1: Solve the absolute value equation \(\left| {2x - 5} \right| = 11\).

Drop the absolute value symbol of the expression on the left and set it to equal to positive and negative \(11\).

$$2x - 5 = {\color{blue}+} 11$$

$$2x - 5 = {\color{red}-} 11$$

Solving the first equation:

$$\eqalign{
2x - 5 &= + 11 \cr
2x &= 11 + 5 \cr
2x &= 16 \cr
x &= 8 \cr} $$

Solving the second equation:

$$\eqalign{
2x - 5 &= - 11 \cr
2x &= - 11 + 5 \cr
2x &= - 6 \cr
x &= - 3 \cr} $$

Therefore, our final answers are

$$\eqalign{
& {x_1} = 8 \cr
& {x_2} = - 3 \cr} $$

Example 2: Solve the absolute value equation \(\left| {3 -{\Large{ {{2x} \over 5}} }}\right| = 9\).

Remove the absolute value symbol and set it equal to positive and negative \(9\).

$$3 - {{2x} \over 5} = {\color{blue}+} 9$$

$$3 - {{2x} \over 5} = {\color{red}-} 9$$

Let's solve the first equation.

$$\eqalign{
3 - {{2x} \over 5} &= + 9 \cr
- {{2x} \over 5} &= 9 - 3 \cr
- {{2x} \over 5} &= 6 \cr
- 2x &= 30 \cr
x &= - 15 \cr} $$

Then the second equation.

$$\eqalign{
3 - {{2x} \over 5} &= - 9 \cr
- {{2x} \over 5} &= - 9 - 3 \cr
- {{2x} \over 5} &= - 12 \cr
- 2x &= - 60 \cr
x &= 30 \cr} $$

Therefore, the final answers are

$$\eqalign{
& {x_1} = - 15 \cr
& {x_2} = 30 \cr} $$

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