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Solution - Absolute value equations

Exact form:

s = 4, - \frac{14}{3}

s=4 , -\frac{14}{3}

Mixed number form:

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

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

Decimal form:

s = 4, - 4.667

s=4 , -4.667

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
$| s + 9 | = | 2 s + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| s + 9 \| = \| 2 s + 5 \|$
$x = + y$	$(s + 9) = (2 s + 5)$
$x = - y$	$(s + 9) = - (2 s + 5)$
$+ x = y$	$(s + 9) = (2 s + 5)$
$- x = y$	$- (s + 9) = (2 s + 5)$

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:

$\| x \| = \| y \|$	$\| s + 9 \| = \| 2 s + 5 \|$
$x = + y$ , $+ x = y$	$(s + 9) = (2 s + 5)$
$x = - y$ , $- x = y$	$(s + 9) = - (2 s + 5)$

2. Solve the two equations for s

10 additional steps

$(s + 9) = (2 s + 5)$

Subtract from both sides:

$(s + 9) - 2 s = (2 s + 5) - 2 s$

Group like terms:

$(s - 2 s) + 9 = (2 s + 5) - 2 s$

Simplify the arithmetic:

$- s + 9 = (2 s + 5) - 2 s$

Group like terms:

$- s + 9 = (2 s - 2 s) + 5$

Simplify the arithmetic:

$- s + 9 = 5$

Subtract from both sides:

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

Simplify the arithmetic:

$- s = 5 - 9$

Simplify the arithmetic:

$- s = - 4$

Multiply both sides by :

$- s \cdot - 1 = - 4 \cdot - 1$

Remove the one(s):

$s = - 4 \cdot - 1$

Simplify the arithmetic:

$s = 4$

10 additional steps

$(s + 9) = - (2 s + 5)$

Expand the parentheses:

$(s + 9) = - 2 s - 5$

Add to both sides:

$(s + 9) + 2 s = (- 2 s - 5) + 2 s$

Group like terms:

$(s + 2 s) + 9 = (- 2 s - 5) + 2 s$

Simplify the arithmetic:

$3 s + 9 = (- 2 s - 5) + 2 s$

Group like terms:

$3 s + 9 = (- 2 s + 2 s) - 5$

Simplify the arithmetic:

$3 s + 9 = - 5$

Subtract from both sides:

$(3 s + 9) - 9 = - 5 - 9$

Simplify the arithmetic:

$3 s = - 5 - 9$

Simplify the arithmetic:

$3 s = - 14$

Divide both sides by :

$\frac{(3 s)}{3} = \frac{- 14}{3}$

Simplify the fraction:

$s = \frac{- 14}{3}$

3. List the solutions

$s = 4, - \frac{14}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | s + 9 |$
$y = | 2 s + 5 |$
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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