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Solution - Absolute value equations

Exact form:

x = - \frac{1}{2}, \frac{3}{2}

x=-\frac{1}{2} , \frac{3}{2}

Mixed number form:

x = - \frac{1}{2}, 1 \frac{1}{2}

x=-\frac{1}{2} , 1\frac{1}{2}

Decimal form:

x = - 0.5, 1.5

x=-0.5 , 1.5

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
$| - 2 x - 3 | = | 4 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 x - 3 \| = \| 4 x \|$
$x = + y$	$(- 2 x - 3) = (4 x)$
$x = - y$	$(- 2 x - 3) = - (4 x)$
$+ x = y$	$(- 2 x - 3) = (4 x)$
$- x = y$	$- (- 2 x - 3) = (4 x)$

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:

$\| x \| = \| y \|$	$\| - 2 x - 3 \| = \| 4 x \|$
$x = + y$ , $+ x = y$	$(- 2 x - 3) = (4 x)$
$x = - y$ , $- x = y$	$(- 2 x - 3) = - (4 x)$

2. Solve the two equations for x

12 additional steps

$(- 2 x - 3) = 4 x$

Subtract from both sides:

$(- 2 x - 3) - 4 x = (4 x) - 4 x$

Group like terms:

$(- 2 x - 4 x) - 3 = (4 x) - 4 x$

Simplify the arithmetic:

$- 6 x - 3 = (4 x) - 4 x$

Simplify the arithmetic:

$- 6 x - 3 = 0$

Add to both sides:

$(- 6 x - 3) + 3 = 0 + 3$

Simplify the arithmetic:

$- 6 x = 0 + 3$

Simplify the arithmetic:

$- 6 x = 3$

Divide both sides by :

$\frac{(- 6 x)}{- 6} = \frac{3}{- 6}$

Cancel out the negatives:

$\frac{6 x}{6} = \frac{3}{- 6}$

Simplify the fraction:

$x = \frac{3}{- 6}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 3}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 3)}{(2 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{2}$

7 additional steps

$(- 2 x - 3) = - 4 x$

Add to both sides:

$(- 2 x - 3) + 3 = (- 4 x) + 3$

Simplify the arithmetic:

$- 2 x = (- 4 x) + 3$

Add to both sides:

$(- 2 x) + 4 x = ((- 4 x) + 3) + 4 x$

Simplify the arithmetic:

$2 x = ((- 4 x) + 3) + 4 x$

Group like terms:

$2 x = (- 4 x + 4 x) + 3$

Simplify the arithmetic:

$2 x = 3$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{3}{2}$

Simplify the fraction:

$x = \frac{3}{2}$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 x - 3 |$
$y = | 4 x |$
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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