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Solution - Absolute value equations

Exact form:

s = - \frac{8}{3}, - \frac{4}{5}

s=-\frac{8}{3} , -\frac{4}{5}

Mixed number form:

s = - 2 \frac{2}{3}, - \frac{4}{5}

s=-2\frac{2}{3} , -\frac{4}{5}

Decimal form:

s = - 2.667, - 0.8

s=-2.667 , -0.8

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
$| 4 s + 6 | = | s - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 s + 6 \| = \| s - 2 \|$
$x = + y$	$(4 s + 6) = (s - 2)$
$x = - y$	$(4 s + 6) = - (s - 2)$
$+ x = y$	$(4 s + 6) = (s - 2)$
$- x = y$	$- (4 s + 6) = (s - 2)$

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:

$\| x \| = \| y \|$	$\| 4 s + 6 \| = \| s - 2 \|$
$x = + y$ , $+ x = y$	$(4 s + 6) = (s - 2)$
$x = - y$ , $- x = y$	$(4 s + 6) = - (s - 2)$

2. Solve the two equations for s

9 additional steps

$(4 s + 6) = (s - 2)$

Subtract from both sides:

$(4 s + 6) - s = (s - 2) - s$

Group like terms:

$(4 s - s) + 6 = (s - 2) - s$

Simplify the arithmetic:

$3 s + 6 = (s - 2) - s$

Group like terms:

$3 s + 6 = (s - s) - 2$

Simplify the arithmetic:

$3 s + 6 = - 2$

Subtract from both sides:

$(3 s + 6) - 6 = - 2 - 6$

Simplify the arithmetic:

$3 s = - 2 - 6$

Simplify the arithmetic:

$3 s = - 8$

Divide both sides by :

$\frac{(3 s)}{3} = \frac{- 8}{3}$

Simplify the fraction:

$s = \frac{- 8}{3}$

10 additional steps

$(4 s + 6) = - (s - 2)$

Expand the parentheses:

$(4 s + 6) = - s + 2$

Add to both sides:

$(4 s + 6) + s = (- s + 2) + s$

Group like terms:

$(4 s + s) + 6 = (- s + 2) + s$

Simplify the arithmetic:

$5 s + 6 = (- s + 2) + s$

Group like terms:

$5 s + 6 = (- s + s) + 2$

Simplify the arithmetic:

$5 s + 6 = 2$

Subtract from both sides:

$(5 s + 6) - 6 = 2 - 6$

Simplify the arithmetic:

$5 s = 2 - 6$

Simplify the arithmetic:

$5 s = - 4$

Divide both sides by :

$\frac{(5 s)}{5} = \frac{- 4}{5}$

Simplify the fraction:

$s = \frac{- 4}{5}$

3. List the solutions

$s = - \frac{8}{3}, - \frac{4}{5}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 s + 6 |$
$y = | s - 2 |$
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 s

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 s

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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