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Solution - Absolute value equations

Exact form:

x = - 5, - 1

x=-5 , -1

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
$- | 5 x + 3 | = - | 6 x + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$- \| 5 x + 3 \| = - \| 6 x + 8 \|$
$x = + y$	$- (5 x + 3) = - (6 x + 8)$
$x = - y$	$- (5 x + 3) = - (- (6 x + 8))$
$+ x = y$	$- (5 x + 3) = - (6 x + 8)$
$- x = y$	$- (- (5 x + 3)) = - (6 x + 8)$

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:

$\| x \| = \| y \|$	$- \| 5 x + 3 \| = - \| 6 x + 8 \|$
$x = + y$ , $+ x = y$	$- (5 x + 3) = - (6 x + 8)$
$x = - y$ , $- x = y$	$- (5 x + 3) = - (- (6 x + 8))$

2. Solve the two equations for x

9 additional steps

$- (5 x + 3) = - (6 x + 8)$

Expand the parentheses:

$- 5 x - 3 = - (6 x + 8)$

Expand the parentheses:

$- 5 x - 3 = - 6 x - 8$

Add to both sides:

$(- 5 x - 3) + 6 x = (- 6 x - 8) + 6 x$

Group like terms:

$(- 5 x + 6 x) - 3 = (- 6 x - 8) + 6 x$

Simplify the arithmetic:

$x - 3 = (- 6 x - 8) + 6 x$

Group like terms:

$x - 3 = (- 6 x + 6 x) - 8$

Simplify the arithmetic:

$x - 3 = - 8$

Add to both sides:

$(x - 3) + 3 = - 8 + 3$

Simplify the arithmetic:

$x = - 8 + 3$

Simplify the arithmetic:

$x = - 5$

14 additional steps

$- (5 x + 3) = - (- (6 x + 8))$

Expand the parentheses:

$- 5 x - 3 = - (- (6 x + 8))$

Resolve the double minus:

$- 5 x - 3 = 6 x + 8$

Subtract from both sides:

$(- 5 x - 3) - 6 x = (6 x + 8) - 6 x$

Group like terms:

$(- 5 x - 6 x) - 3 = (6 x + 8) - 6 x$

Simplify the arithmetic:

$- 11 x - 3 = (6 x + 8) - 6 x$

Group like terms:

$- 11 x - 3 = (6 x - 6 x) + 8$

Simplify the arithmetic:

$- 11 x - 3 = 8$

Add to both sides:

$(- 11 x - 3) + 3 = 8 + 3$

Simplify the arithmetic:

$- 11 x = 8 + 3$

Simplify the arithmetic:

$- 11 x = 11$

Divide both sides by :

$\frac{(- 11 x)}{- 11} = \frac{11}{- 11}$

Cancel out the negatives:

$\frac{11 x}{11} = \frac{11}{- 11}$

Simplify the fraction:

$x = \frac{11}{- 11}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 11}{11}$

Simplify the fraction:

$x = - 1$

3. List the solutions

$x = - 5, - 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = - | 5 x + 3 |$
$y = - | 6 x + 8 |$
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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