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Solution - Absolute value equations

Exact form:

x = - \frac{1}{7}, - \frac{23}{3}

x=-\frac{1}{7} , -\frac{23}{3}

Mixed number form:

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

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

Decimal form:

x = - 0.143, - 7.667

x=-0.143 , -7.667

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
$| 5 x + 12 | = | - 2 x + 11 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 5 x + 12 \| = \| - 2 x + 11 \|$
$x = + y$	$(5 x + 12) = (- 2 x + 11)$
$x = - y$	$(5 x + 12) = - (- 2 x + 11)$
$+ x = y$	$(5 x + 12) = (- 2 x + 11)$
$- x = y$	$- (5 x + 12) = (- 2 x + 11)$

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:

$\| x \| = \| y \|$	$\| 5 x + 12 \| = \| - 2 x + 11 \|$
$x = + y$ , $+ x = y$	$(5 x + 12) = (- 2 x + 11)$
$x = - y$ , $- x = y$	$(5 x + 12) = - (- 2 x + 11)$

2. Solve the two equations for x

9 additional steps

$(5 x + 12) = (- 2 x + 11)$

Add to both sides:

$(5 x + 12) + 2 x = (- 2 x + 11) + 2 x$

Group like terms:

$(5 x + 2 x) + 12 = (- 2 x + 11) + 2 x$

Simplify the arithmetic:

$7 x + 12 = (- 2 x + 11) + 2 x$

Group like terms:

$7 x + 12 = (- 2 x + 2 x) + 11$

Simplify the arithmetic:

$7 x + 12 = 11$

Subtract from both sides:

$(7 x + 12) - 12 = 11 - 12$

Simplify the arithmetic:

$7 x = 11 - 12$

Simplify the arithmetic:

$7 x = - 1$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{- 1}{7}$

Simplify the fraction:

$x = \frac{- 1}{7}$

10 additional steps

$(5 x + 12) = - (- 2 x + 11)$

Expand the parentheses:

$(5 x + 12) = 2 x - 11$

Subtract from both sides:

$(5 x + 12) - 2 x = (2 x - 11) - 2 x$

Group like terms:

$(5 x - 2 x) + 12 = (2 x - 11) - 2 x$

Simplify the arithmetic:

$3 x + 12 = (2 x - 11) - 2 x$

Group like terms:

$3 x + 12 = (2 x - 2 x) - 11$

Simplify the arithmetic:

$3 x + 12 = - 11$

Subtract from both sides:

$(3 x + 12) - 12 = - 11 - 12$

Simplify the arithmetic:

$3 x = - 11 - 12$

Simplify the arithmetic:

$3 x = - 23$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{- 23}{3}$

Simplify the fraction:

$x = \frac{- 23}{3}$

3. List the solutions

$x = - \frac{1}{7}, - \frac{23}{3}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 5 x + 12 |$
$y = | - 2 x + 11 |$
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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