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Solution - Absolute value equations

Exact form:

x = \frac{5}{8}

x=\frac{5}{8}

Decimal form:

x = 0.625

x=0.625

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 4 x - 5 | + | 4 x | = 0$

Add $- | 4 x |$ to both sides of the equation:

$| 4 x - 5 | + | 4 x | - | 4 x | = - | 4 x |$

Simplify the arithmetic

$| 4 x - 5 | = - | 4 x |$

2. 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
$| 4 x - 5 | = - | 4 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 5 \| = - \| 4 x \|$
$x = + y$	$(4 x - 5) = - (4 x)$
$x = - y$	$(4 x - 5) = - - (4 x)$
$+ x = y$	$(4 x - 5) = - (4 x)$
$- x = y$	$- (4 x - 5) = - (4 x)$

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:

$\| x \| = \| y \|$	$\| 4 x - 5 \| = - \| 4 x \|$
$x = + y$ , $+ x = y$	$(4 x - 5) = - (4 x)$
$x = - y$ , $- x = y$	$(4 x - 5) = - - (4 x)$

3. Solve the two equations for x

7 additional steps

$(4 x - 5) = - 4 x$

Add to both sides:

$(4 x - 5) + 5 = (- 4 x) + 5$

Simplify the arithmetic:

$4 x = (- 4 x) + 5$

Add to both sides:

$(4 x) + 4 x = ((- 4 x) + 5) + 4 x$

Simplify the arithmetic:

$8 x = ((- 4 x) + 5) + 4 x$

Group like terms:

$8 x = (- 4 x + 4 x) + 5$

Simplify the arithmetic:

$8 x = 5$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{5}{8}$

Simplify the fraction:

$x = \frac{5}{8}$

6 additional steps

$(4 x - 5) = - - 4 x$

Group like terms:

$(4 x - 5) = (- 1 \cdot - 4) x$

Multiply the coefficients:

$(4 x - 5) = 4 x$

Subtract from both sides:

$(4 x - 5) - 4 x = (4 x) - 4 x$

Group like terms:

$(4 x - 4 x) - 5 = (4 x) - 4 x$

Simplify the arithmetic:

$- 5 = (4 x) - 4 x$

Simplify the arithmetic:

$- 5 = 0$

The statement is false:

$- 5 = 0$

The equation is false so it has no solution.

4. List the solutions

$x = \frac{5}{8}$
(1 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 5 |$
$y = - | 4 x |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

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