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Solution - Absolute value equations

Exact form:

x = \frac{1}{3}, - \frac{5}{7}

x=\frac{1}{3} , -\frac{5}{7}

Decimal form:

x = 0.333, - 0.714

x=0.333 , -0.714

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
$| 12 x + 7 | = | 9 x + 8 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 12 x + 7 \| = \| 9 x + 8 \|$
$x = + y$	$(12 x + 7) = (9 x + 8)$
$x = - y$	$(12 x + 7) = - (9 x + 8)$
$+ x = y$	$(12 x + 7) = (9 x + 8)$
$- x = y$	$- (12 x + 7) = (9 x + 8)$

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:

$\| x \| = \| y \|$	$\| 12 x + 7 \| = \| 9 x + 8 \|$
$x = + y$ , $+ x = y$	$(12 x + 7) = (9 x + 8)$
$x = - y$ , $- x = y$	$(12 x + 7) = - (9 x + 8)$

2. Solve the two equations for x

9 additional steps

$(12 x + 7) = (9 x + 8)$

Subtract from both sides:

$(12 x + 7) - 9 x = (9 x + 8) - 9 x$

Group like terms:

$(12 x - 9 x) + 7 = (9 x + 8) - 9 x$

Simplify the arithmetic:

$3 x + 7 = (9 x + 8) - 9 x$

Group like terms:

$3 x + 7 = (9 x - 9 x) + 8$

Simplify the arithmetic:

$3 x + 7 = 8$

Subtract from both sides:

$(3 x + 7) - 7 = 8 - 7$

Simplify the arithmetic:

$3 x = 8 - 7$

Simplify the arithmetic:

$3 x = 1$

Divide both sides by :

$\frac{(3 x)}{3} = \frac{1}{3}$

Simplify the fraction:

$x = \frac{1}{3}$

12 additional steps

$(12 x + 7) = - (9 x + 8)$

Expand the parentheses:

$(12 x + 7) = - 9 x - 8$

Add to both sides:

$(12 x + 7) + 9 x = (- 9 x - 8) + 9 x$

Group like terms:

$(12 x + 9 x) + 7 = (- 9 x - 8) + 9 x$

Simplify the arithmetic:

$21 x + 7 = (- 9 x - 8) + 9 x$

Group like terms:

$21 x + 7 = (- 9 x + 9 x) - 8$

Simplify the arithmetic:

$21 x + 7 = - 8$

Subtract from both sides:

$(21 x + 7) - 7 = - 8 - 7$

Simplify the arithmetic:

$21 x = - 8 - 7$

Simplify the arithmetic:

$21 x = - 15$

Divide both sides by :

$\frac{(21 x)}{21} = \frac{- 15}{21}$

Simplify the fraction:

$x = \frac{- 15}{21}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 3)}{(7 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 5}{7}$

3. List the solutions

$x = \frac{1}{3}, - \frac{5}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 12 x + 7 |$
$y = | 9 x + 8 |$
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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