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Solution - Absolute value equations

Exact form:

x = 1, \frac{3}{5}

x=1 , \frac{3}{5}

Decimal form:

x = 1, 0.6

x=1 , 0.6

See steps

Step-by-step explanation

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

$| x | - | 4 x - 3 | = 0$

Add $| 4 x - 3 |$ to both sides of the equation:

$| x | - | 4 x - 3 | + | 4 x - 3 | = | 4 x - 3 |$

Simplify the arithmetic

$| x | = | 4 x - 3 |$

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
$| x | = | 4 x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| x \| = \| 4 x - 3 \|$
$x = + y$	$(x) = (4 x - 3)$
$x = - y$	$(x) = (- (4 x - 3))$
$+ x = y$	$(x) = (4 x - 3)$
$- x = y$	$- (x) = (4 x - 3)$

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:

$\| x \| = \| y \|$	$\| x \| = \| 4 x - 3 \|$
$x = + y$ , $+ x = y$	$(x) = (4 x - 3)$
$x = - y$ , $- x = y$	$(x) = (- (4 x - 3))$

3. Solve the two equations for x

8 additional steps

$x = (4 x - 3)$

Subtract from both sides:

$x - 4 x = (4 x - 3) - 4 x$

Simplify the arithmetic:

$- 3 x = (4 x - 3) - 4 x$

Group like terms:

$- 3 x = (4 x - 4 x) - 3$

Simplify the arithmetic:

$- 3 x = - 3$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{- 3}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{- 3}{- 3}$

Simplify the fraction:

$x = \frac{- 3}{- 3}$

Cancel out the negatives:

$x = \frac{3}{3}$

Simplify the fraction:

$x = 1$

6 additional steps

$x = - (4 x - 3)$

Expand the parentheses:

$x = - 4 x + 3$

Add to both sides:

$x + 4 x = (- 4 x + 3) + 4 x$

Simplify the arithmetic:

$5 x = (- 4 x + 3) + 4 x$

Group like terms:

$5 x = (- 4 x + 4 x) + 3$

Simplify the arithmetic:

$5 x = 3$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{3}{5}$

Simplify the fraction:

$x = \frac{3}{5}$

4. List the solutions

$x = 1, \frac{3}{5}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | x |$
$y = | 4 x - 3 |$
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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