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Solution - Absolute value equations

Exact form:

k = - \frac{7}{8}, \frac{3}{2}

k=-\frac{7}{8} , \frac{3}{2}

Mixed number form:

k = - \frac{7}{8}, 1 \frac{1}{2}

k=-\frac{7}{8} , 1\frac{1}{2}

Decimal form:

k = - 0.875, 1.5

k=-0.875 , 1.5

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
$| - 3 k - 5 | = | 5 k + 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 k - 5 \| = \| 5 k + 2 \|$
$x = + y$	$(- 3 k - 5) = (5 k + 2)$
$x = - y$	$(- 3 k - 5) = - (5 k + 2)$
$+ x = y$	$(- 3 k - 5) = (5 k + 2)$
$- x = y$	$- (- 3 k - 5) = (5 k + 2)$

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:

$\| x \| = \| y \|$	$\| - 3 k - 5 \| = \| 5 k + 2 \|$
$x = + y$ , $+ x = y$	$(- 3 k - 5) = (5 k + 2)$
$x = - y$ , $- x = y$	$(- 3 k - 5) = - (5 k + 2)$

2. Solve the two equations for k

11 additional steps

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

$- 8 k - 5 = (5 k + 2) - 5 k$

Group like terms:

$- 8 k - 5 = (5 k - 5 k) + 2$

Simplify the arithmetic:

$- 8 k - 5 = 2$

Add to both sides:

$(- 8 k - 5) + 5 = 2 + 5$

Simplify the arithmetic:

$- 8 k = 2 + 5$

Simplify the arithmetic:

$- 8 k = 7$

Divide both sides by :

$\frac{(- 8 k)}{- 8} = \frac{7}{- 8}$

Cancel out the negatives:

$\frac{8 k}{8} = \frac{7}{- 8}$

Simplify the fraction:

$k = \frac{7}{- 8}$

Move the negative sign from the denominator to the numerator:

$k = \frac{- 7}{8}$

10 additional steps

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

Expand the parentheses:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 k - 5 = - 2$

Add to both sides:

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

Simplify the arithmetic:

$2 k = - 2 + 5$

Simplify the arithmetic:

$2 k = 3$

Divide both sides by :

$\frac{(2 k)}{2} = \frac{3}{2}$

Simplify the fraction:

$k = \frac{3}{2}$

3. List the solutions

$k = - \frac{7}{8}, \frac{3}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 3 k - 5 |$
$y = | 5 k + 2 |$
The equation is true where the two lines cross.

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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

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