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Solution - Absolute value equations

Exact form:

x = \frac{19}{4}, \frac{19}{12}

x=\frac{19}{4} , \frac{19}{12}

Mixed number form:

x = 4 \frac{3}{4}, 1 \frac{7}{12}

x=4\frac{3}{4} , 1\frac{7}{12}

Decimal form:

x = 4.75, 1.583

x=4.75 , 1.583

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
$| 8 x - 19 | = | 4 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 x - 19 \| = \| 4 x \|$
$x = + y$	$(8 x - 19) = (4 x)$
$x = - y$	$(8 x - 19) = - (4 x)$
$+ x = y$	$(8 x - 19) = (4 x)$
$- x = y$	$- (8 x - 19) = (4 x)$

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:

$\| x \| = \| y \|$	$\| 8 x - 19 \| = \| 4 x \|$
$x = + y$ , $+ x = y$	$(8 x - 19) = (4 x)$
$x = - y$ , $- x = y$	$(8 x - 19) = - (4 x)$

2. Solve the two equations for x

8 additional steps

$(8 x - 19) = 4 x$

Subtract from both sides:

$(8 x - 19) - 4 x = (4 x) - 4 x$

Group like terms:

$(8 x - 4 x) - 19 = (4 x) - 4 x$

Simplify the arithmetic:

$4 x - 19 = (4 x) - 4 x$

Simplify the arithmetic:

$4 x - 19 = 0$

Add to both sides:

$(4 x - 19) + 19 = 0 + 19$

Simplify the arithmetic:

$4 x = 0 + 19$

Simplify the arithmetic:

$4 x = 19$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{19}{4}$

Simplify the fraction:

$x = \frac{19}{4}$

7 additional steps

$(8 x - 19) = - 4 x$

Add to both sides:

$(8 x - 19) + 19 = (- 4 x) + 19$

Simplify the arithmetic:

$8 x = (- 4 x) + 19$

Add to both sides:

$(8 x) + 4 x = ((- 4 x) + 19) + 4 x$

Simplify the arithmetic:

$12 x = ((- 4 x) + 19) + 4 x$

Group like terms:

$12 x = (- 4 x + 4 x) + 19$

Simplify the arithmetic:

$12 x = 19$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{19}{12}$

Simplify the fraction:

$x = \frac{19}{12}$

3. List the solutions

$x = \frac{19}{4}, \frac{19}{12}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 8 x - 19 |$
$y = | 4 x |$
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. 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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

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