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Solution - Absolute value equations

Exact form:

s = - \frac{9}{4}, \frac{3}{2}

s=-\frac{9}{4} , \frac{3}{2}

Mixed number form:

s = - 2 \frac{1}{4}, 1 \frac{1}{2}

s=-2\frac{1}{4} , 1\frac{1}{2}

Decimal form:

s = - 2.25, 1.5

s=-2.25 , 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
$| 5 s | = | s - 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 s \| = \| s - 9 \|$
$x = + y$	$(5 s) = (s - 9)$
$x = - y$	$(5 s) = - (s - 9)$
$+ x = y$	$(5 s) = (s - 9)$
$- x = y$	$- (5 s) = (s - 9)$

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:

$\| x \| = \| y \|$	$\| 5 s \| = \| s - 9 \|$
$x = + y$ , $+ x = y$	$(5 s) = (s - 9)$
$x = - y$ , $- x = y$	$(5 s) = - (s - 9)$

2. Solve the two equations for s

5 additional steps

$5 s = (s - 9)$

Subtract from both sides:

$(5 s) - s = (s - 9) - s$

Simplify the arithmetic:

$4 s = (s - 9) - s$

Group like terms:

$4 s = (s - s) - 9$

Simplify the arithmetic:

$4 s = - 9$

Divide both sides by :

$\frac{(4 s)}{4} = \frac{- 9}{4}$

Simplify the fraction:

$s = \frac{- 9}{4}$

8 additional steps

$5 s = - (s - 9)$

Expand the parentheses:

$5 s = - s + 9$

Add to both sides:

$(5 s) + s = (- s + 9) + s$

Simplify the arithmetic:

$6 s = (- s + 9) + s$

Group like terms:

$6 s = (- s + s) + 9$

Simplify the arithmetic:

$6 s = 9$

Divide both sides by :

$\frac{(6 s)}{6} = \frac{9}{6}$

Simplify the fraction:

$s = \frac{9}{6}$

Find the greatest common factor of the numerator and denominator:

$s = \frac{(3 \cdot 3)}{(2 \cdot 3)}$

Factor out and cancel the greatest common factor:

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

3. List the solutions

$s = - \frac{9}{4}, \frac{3}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 s |$
$y = | s - 9 |$
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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