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Solution - Absolute value equations

Exact form:

s = \frac{4}{7}, \frac{10}{9}

s=\frac{4}{7} , \frac{10}{9}

Mixed number form:

s = \frac{4}{7}, 1 \frac{1}{9}

s=\frac{4}{7} , 1\frac{1}{9}

Decimal form:

s = 0.571, 1.111

s=0.571 , 1.111

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
$| 8 s - 7 | = | s - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 s - 7 \| = \| s - 3 \|$
$x = + y$	$(8 s - 7) = (s - 3)$
$x = - y$	$(8 s - 7) = - (s - 3)$
$+ x = y$	$(8 s - 7) = (s - 3)$
$- x = y$	$- (8 s - 7) = (s - 3)$

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:

$\| x \| = \| y \|$	$\| 8 s - 7 \| = \| s - 3 \|$
$x = + y$ , $+ x = y$	$(8 s - 7) = (s - 3)$
$x = - y$ , $- x = y$	$(8 s - 7) = - (s - 3)$

2. Solve the two equations for s

9 additional steps

$(8 s - 7) = (s - 3)$

Subtract from both sides:

$(8 s - 7) - s = (s - 3) - s$

Group like terms:

$(8 s - s) - 7 = (s - 3) - s$

Simplify the arithmetic:

$7 s - 7 = (s - 3) - s$

Group like terms:

$7 s - 7 = (s - s) - 3$

Simplify the arithmetic:

$7 s - 7 = - 3$

Add to both sides:

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

Simplify the arithmetic:

$7 s = - 3 + 7$

Simplify the arithmetic:

$7 s = 4$

Divide both sides by :

$\frac{(7 s)}{7} = \frac{4}{7}$

Simplify the fraction:

$s = \frac{4}{7}$

10 additional steps

$(8 s - 7) = - (s - 3)$

Expand the parentheses:

$(8 s - 7) = - s + 3$

Add to both sides:

$(8 s - 7) + s = (- s + 3) + s$

Group like terms:

$(8 s + s) - 7 = (- s + 3) + s$

Simplify the arithmetic:

$9 s - 7 = (- s + 3) + s$

Group like terms:

$9 s - 7 = (- s + s) + 3$

Simplify the arithmetic:

$9 s - 7 = 3$

Add to both sides:

$(9 s - 7) + 7 = 3 + 7$

Simplify the arithmetic:

$9 s = 3 + 7$

Simplify the arithmetic:

$9 s = 10$

Divide both sides by :

$\frac{(9 s)}{9} = \frac{10}{9}$

Simplify the fraction:

$s = \frac{10}{9}$

3. List the solutions

$s = \frac{4}{7}, \frac{10}{9}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 8 s - 7 |$
$y = | s - 3 |$
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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