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Solution - Absolute value equations

Exact form:

x = \frac{5}{16}

x=\frac{5}{16}

Decimal form:

x = 0.312

x=0.312

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
$| 8 x | = | 8 x - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 x \| = \| 8 x - 5 \|$
$x = + y$	$(8 x) = (8 x - 5)$
$x = - y$	$(8 x) = - (8 x - 5)$
$+ x = y$	$(8 x) = (8 x - 5)$
$- x = y$	$- (8 x) = (8 x - 5)$

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:

$\| x \| = \| y \|$	$\| 8 x \| = \| 8 x - 5 \|$
$x = + y$ , $+ x = y$	$(8 x) = (8 x - 5)$
$x = - y$ , $- x = y$	$(8 x) = - (8 x - 5)$

2. Solve the two equations for x

4 additional steps

$8 x = (8 x - 5)$

Subtract from both sides:

$(8 x) - 8 x = (8 x - 5) - 8 x$

Simplify the arithmetic:

$0 = (8 x - 5) - 8 x$

Group like terms:

$0 = (8 x - 8 x) - 5$

Simplify the arithmetic:

$0 = - 5$

The statement is false:

$0 = - 5$

The equation is false so it has no solution.

6 additional steps

$8 x = - (8 x - 5)$

Expand the parentheses:

$8 x = - 8 x + 5$

Add to both sides:

$(8 x) + 8 x = (- 8 x + 5) + 8 x$

Simplify the arithmetic:

$16 x = (- 8 x + 5) + 8 x$

Group like terms:

$16 x = (- 8 x + 8 x) + 5$

Simplify the arithmetic:

$16 x = 5$

Divide both sides by :

$\frac{(16 x)}{16} = \frac{5}{16}$

Simplify the fraction:

$x = \frac{5}{16}$

3. Graph

Each line represents the function of one side of the equation:
$y = | 8 x |$
$y = | 8 x - 5 |$
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 x

3. 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 x

3. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved