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Solution - Absolute value equations

Exact form:

s = \frac{11}{2}, \frac{11}{4}

s=\frac{11}{2} , \frac{11}{4}

Mixed number form:

s = 5 \frac{1}{2}, 2 \frac{3}{4}

s=5\frac{1}{2} , 2\frac{3}{4}

Decimal form:

s = 5.5, 2.75

s=5.5 , 2.75

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
$| 3 s - 11 | = | s |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 s - 11 \| = \| s \|$
$x = + y$	$(3 s - 11) = (s)$
$x = - y$	$(3 s - 11) = - (s)$
$+ x = y$	$(3 s - 11) = (s)$
$- x = y$	$- (3 s - 11) = (s)$

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:

$\| x \| = \| y \|$	$\| 3 s - 11 \| = \| s \|$
$x = + y$ , $+ x = y$	$(3 s - 11) = (s)$
$x = - y$ , $- x = y$	$(3 s - 11) = - (s)$

2. Solve the two equations for s

8 additional steps

$(3 s - 11) = s$

Subtract from both sides:

$(3 s - 11) - s = s - s$

Group like terms:

$(3 s - s) - 11 = s - s$

Simplify the arithmetic:

$2 s - 11 = s - s$

Simplify the arithmetic:

$2 s - 11 = 0$

Add to both sides:

$(2 s - 11) + 11 = 0 + 11$

Simplify the arithmetic:

$2 s = 0 + 11$

Simplify the arithmetic:

$2 s = 11$

Divide both sides by :

$\frac{(2 s)}{2} = \frac{11}{2}$

Simplify the fraction:

$s = \frac{11}{2}$

8 additional steps

$(3 s - 11) = - s$

Add to both sides:

$(3 s - 11) + s = - s + s$

Group like terms:

$(3 s + s) - 11 = - s + s$

Simplify the arithmetic:

$4 s - 11 = - s + s$

Simplify the arithmetic:

$4 s - 11 = 0$

Add to both sides:

$(4 s - 11) + 11 = 0 + 11$

Simplify the arithmetic:

$4 s = 0 + 11$

Simplify the arithmetic:

$4 s = 11$

Divide both sides by :

$\frac{(4 s)}{4} = \frac{11}{4}$

Simplify the fraction:

$s = \frac{11}{4}$

3. List the solutions

$s = \frac{11}{2}, \frac{11}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 s - 11 |$
$y = | s |$
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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