Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

k = - 1

k=-1

See steps

Step-by-step explanation

1. Rewrite the equation without absolute value bars

Use the rules:
$| x | = | y |$ → $x = \pm y$ and $| x | = | y |$ → $\pm x = y$
to write all four options of the equation
$| k - 8 | = | k + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| k - 8 \| = \| k + 10 \|$
$x = + y$	$(k - 8) = (k + 10)$
$x = - y$	$(k - 8) = - (k + 10)$
$+ x = y$	$(k - 8) = (k + 10)$
$- x = y$	$- (k - 8) = (k + 10)$

When simplified, equations $x = + y$ and $+ x = y$ are the same and equations $x = - y$ and $- x = y$ are the same, so we end up with only 2 equations:

$\| x \| = \| y \|$	$\| k - 8 \| = \| k + 10 \|$
$x = + y$ , $+ x = y$	$(k - 8) = (k + 10)$
$x = - y$ , $- x = y$	$(k - 8) = - (k + 10)$

2. Solve the two equations for k

5 additional steps

$(k - 8) = (k + 10)$

Subtract from both sides:

$(k - 8) - k = (k + 10) - k$

Group like terms:

$(k - k) - 8 = (k + 10) - k$

Simplify the arithmetic:

$- 8 = (k + 10) - k$

Group like terms:

$- 8 = (k - k) + 10$

Simplify the arithmetic:

$- 8 = 10$

The statement is false:

$- 8 = 10$

The equation is false so it has no solution.

11 additional steps

$(k - 8) = - (k + 10)$

Expand the parentheses:

$(k - 8) = - k - 10$

Add to both sides:

$(k - 8) + k = (- k - 10) + k$

Group like terms:

$(k + k) - 8 = (- k - 10) + k$

Simplify the arithmetic:

$2 k - 8 = (- k - 10) + k$

Group like terms:

$2 k - 8 = (- k + k) - 10$

Simplify the arithmetic:

$2 k - 8 = - 10$

Add to both sides:

$(2 k - 8) + 8 = - 10 + 8$

Simplify the arithmetic:

$2 k = - 10 + 8$

Simplify the arithmetic:

$2 k = - 2$

Divide both sides by :

$\frac{(2 k)}{2} = \frac{- 2}{2}$

Simplify the fraction:

$k = \frac{- 2}{2}$

Simplify the fraction:

$k = - 1$

3. Graph

Each line represents the function of one side of the equation:
$y = | k - 8 |$
$y = | k + 10 |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

Why learn this

We encounter absolute values almost daily. For example: If you walk 3 miles to school, do you also walk minus 3 miles when you go back home? The answer is no because distances use absolute value. The absolute value of the distance between home and school is 3 miles, there or back.
In short, absolute values help us deal with concepts like distance, ranges of possible values, and deviation from a set value.

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for k

3. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation without absolute value bars

2. Solve the two equations for k

3. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved