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Solution - Absolute value equations

Exact form:

x = - \frac{5}{2}, - \frac{5}{4}

x=-\frac{5}{2} , -\frac{5}{4}

Mixed number form:

x = - 2 \frac{1}{2}, - 1 \frac{1}{4}

x=-2\frac{1}{2} , -1\frac{1}{4}

Decimal form:

x = - 2.5, - 1.25

x=-2.5 , -1.25

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
$2 | 3 x + 5 | = | 2 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$2 \| 3 x + 5 \| = \| 2 x \|$
$x = + y$	$2 (3 x + 5) = (2 x)$
$x = - y$	$2 (3 x + 5) = - (2 x)$
$+ x = y$	$2 (3 x + 5) = (2 x)$
$- x = y$	$2 (- (3 x + 5)) = (2 x)$

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:

$\| x \| = \| y \|$	$2 \| 3 x + 5 \| = \| 2 x \|$
$x = + y$ , $+ x = y$	$2 (3 x + 5) = (2 x)$
$x = - y$ , $- x = y$	$2 (3 x + 5) = - (2 x)$

2. Solve the two equations for x

13 additional steps

$2 \cdot (3 x + 5) = 2 x$

Expand the parentheses:

$2 \cdot 3 x + 2 \cdot 5 = 2 x$

Multiply the coefficients:

$6 x + 2 \cdot 5 = 2 x$

Simplify the arithmetic:

$6 x + 10 = 2 x$

Subtract from both sides:

$(6 x + 10) - 2 x = (2 x) - 2 x$

Group like terms:

$(6 x - 2 x) + 10 = (2 x) - 2 x$

Simplify the arithmetic:

$4 x + 10 = (2 x) - 2 x$

Simplify the arithmetic:

$4 x + 10 = 0$

Subtract from both sides:

$(4 x + 10) - 10 = 0 - 10$

Simplify the arithmetic:

$4 x = 0 - 10$

Simplify the arithmetic:

$4 x = - 10$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{- 10}{4}$

Simplify the fraction:

$x = \frac{- 10}{4}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 5}{2}$

13 additional steps

$2 \cdot (3 x + 5) = - (2 x)$

Expand the parentheses:

$2 \cdot 3 x + 2 \cdot 5 = - (2 x)$

Multiply the coefficients:

$6 x + 2 \cdot 5 = - (2 x)$

Simplify the arithmetic:

$6 x + 10 = - (2 x)$

Add to both sides:

$(6 x + 10) + 2 x = (- 2 x) + 2 x$

Group like terms:

$(6 x + 2 x) + 10 = (- 2 x) + 2 x$

Simplify the arithmetic:

$8 x + 10 = (- 2 x) + 2 x$

Simplify the arithmetic:

$8 x + 10 = 0$

Subtract from both sides:

$(8 x + 10) - 10 = 0 - 10$

Simplify the arithmetic:

$8 x = 0 - 10$

Simplify the arithmetic:

$8 x = - 10$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{- 10}{8}$

Simplify the fraction:

$x = \frac{- 10}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 2)}{(4 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 5}{4}$

3. List the solutions

$x = - \frac{5}{2}, - \frac{5}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 2 | 3 x + 5 |$
$y = | 2 x |$
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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