Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

k = 2, - 4

k=2 , -4

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
$| - 3 k + 3 | = | 2 k - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 k + 3 \| = \| 2 k - 7 \|$
$x = + y$	$(- 3 k + 3) = (2 k - 7)$
$x = - y$	$(- 3 k + 3) = - (2 k - 7)$
$+ x = y$	$(- 3 k + 3) = (2 k - 7)$
$- x = y$	$- (- 3 k + 3) = (2 k - 7)$

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:

$\| x \| = \| y \|$	$\| - 3 k + 3 \| = \| 2 k - 7 \|$
$x = + y$ , $+ x = y$	$(- 3 k + 3) = (2 k - 7)$
$x = - y$ , $- x = y$	$(- 3 k + 3) = - (2 k - 7)$

2. Solve the two equations for k

13 additional steps

$(- 3 k + 3) = (2 k - 7)$

Subtract from both sides:

$(- 3 k + 3) - 2 k = (2 k - 7) - 2 k$

Group like terms:

$(- 3 k - 2 k) + 3 = (2 k - 7) - 2 k$

Simplify the arithmetic:

$- 5 k + 3 = (2 k - 7) - 2 k$

Group like terms:

$- 5 k + 3 = (2 k - 2 k) - 7$

Simplify the arithmetic:

$- 5 k + 3 = - 7$

Subtract from both sides:

$(- 5 k + 3) - 3 = - 7 - 3$

Simplify the arithmetic:

$- 5 k = - 7 - 3$

Simplify the arithmetic:

$- 5 k = - 10$

Divide both sides by :

$\frac{(- 5 k)}{- 5} = \frac{- 10}{- 5}$

Cancel out the negatives:

$\frac{5 k}{5} = \frac{- 10}{- 5}$

Simplify the fraction:

$k = \frac{- 10}{- 5}$

Cancel out the negatives:

$k = \frac{10}{5}$

Find the greatest common factor of the numerator and denominator:

$k = \frac{(2 \cdot 5)}{(1 \cdot 5)}$

Factor out and cancel the greatest common factor:

$k = 2$

11 additional steps

$(- 3 k + 3) = - (2 k - 7)$

Expand the parentheses:

$(- 3 k + 3) = - 2 k + 7$

Add to both sides:

$(- 3 k + 3) + 2 k = (- 2 k + 7) + 2 k$

Group like terms:

$(- 3 k + 2 k) + 3 = (- 2 k + 7) + 2 k$

Simplify the arithmetic:

$- k + 3 = (- 2 k + 7) + 2 k$

Group like terms:

$- k + 3 = (- 2 k + 2 k) + 7$

Simplify the arithmetic:

$- k + 3 = 7$

Subtract from both sides:

$(- k + 3) - 3 = 7 - 3$

Simplify the arithmetic:

$- k = 7 - 3$

Simplify the arithmetic:

$- k = 4$

Multiply both sides by :

$- k \cdot - 1 = 4 \cdot - 1$

Remove the one(s):

$k = 4 \cdot - 1$

Simplify the arithmetic:

$k = - 4$

3. List the solutions

$k = 2, - 4$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 3 k + 3 |$
$y = | 2 k - 7 |$
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. List the solutions

4. 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. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved