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Solution - Absolute value equations

Exact form:

x = - \frac{3}{7}, \frac{7}{3}

x=-\frac{3}{7} , \frac{7}{3}

Mixed number form:

x = - \frac{3}{7}, 2 \frac{1}{3}

x=-\frac{3}{7} , 2\frac{1}{3}

Decimal form:

x = - 0.429, 2.333

x=-0.429 , 2.333

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

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

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:

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

2. Solve the two equations for x

9 additional steps

$(2 x + 5) = (- 5 x + 2)$

Add to both sides:

$(2 x + 5) + 5 x = (- 5 x + 2) + 5 x$

Group like terms:

$(2 x + 5 x) + 5 = (- 5 x + 2) + 5 x$

Simplify the arithmetic:

$7 x + 5 = (- 5 x + 2) + 5 x$

Group like terms:

$7 x + 5 = (- 5 x + 5 x) + 2$

Simplify the arithmetic:

$7 x + 5 = 2$

Subtract from both sides:

$(7 x + 5) - 5 = 2 - 5$

Simplify the arithmetic:

$7 x = 2 - 5$

Simplify the arithmetic:

$7 x = - 3$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{- 3}{7}$

Simplify the fraction:

$x = \frac{- 3}{7}$

12 additional steps

$(2 x + 5) = - (- 5 x + 2)$

Expand the parentheses:

$(2 x + 5) = 5 x - 2$

Subtract from both sides:

$(2 x + 5) - 5 x = (5 x - 2) - 5 x$

Group like terms:

$(2 x - 5 x) + 5 = (5 x - 2) - 5 x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 3 x + 5 = - 2$

Subtract from both sides:

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

Simplify the arithmetic:

$- 3 x = - 2 - 5$

Simplify the arithmetic:

$- 3 x = - 7$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{- 7}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{- 7}{- 3}$

Simplify the fraction:

$x = \frac{- 7}{- 3}$

Cancel out the negatives:

$x = \frac{7}{3}$

3. List the solutions

$x = - \frac{3}{7}, \frac{7}{3}$
(2 solution(s))

4. Graph

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