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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

s = - 1, 0.5

s=-1 , 0.5

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
$| 3 s | = | s - 2 |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for s

6 additional steps

$3 s = (s - 2)$

Subtract from both sides:

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

Simplify the arithmetic:

$2 s = (s - 2) - s$

Group like terms:

$2 s = (s - s) - 2$

Simplify the arithmetic:

$2 s = - 2$

Divide both sides by :

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

Simplify the fraction:

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

Simplify the fraction:

$s = - 1$

8 additional steps

$3 s = - (s - 2)$

Expand the parentheses:

$3 s = - s + 2$

Add to both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$4 s = 2$

Divide both sides by :

$\frac{(4 s)}{4} = \frac{2}{4}$

Simplify the fraction:

$s = \frac{2}{4}$

Find the greatest common factor of the numerator and denominator:

$s = \frac{(1 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$s = \frac{1}{2}$

3. List the solutions

$s = - 1, \frac{1}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 s |$
$y = | s - 2 |$
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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