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Solution - Absolute value equations

Exact form:

x = - 6, \frac{1}{4}

x=-6 , \frac{1}{4}

Decimal form:

x = - 6, 0.25

x=-6 , 0.25

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
$| 7 x - 8 | = | 9 x + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 x - 8 \| = \| 9 x + 4 \|$
$x = + y$	$(7 x - 8) = (9 x + 4)$
$x = - y$	$(7 x - 8) = - (9 x + 4)$
$+ x = y$	$(7 x - 8) = (9 x + 4)$
$- x = y$	$- (7 x - 8) = (9 x + 4)$

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:

$\| x \| = \| y \|$	$\| 7 x - 8 \| = \| 9 x + 4 \|$
$x = + y$ , $+ x = y$	$(7 x - 8) = (9 x + 4)$
$x = - y$ , $- x = y$	$(7 x - 8) = - (9 x + 4)$

2. Solve the two equations for x

13 additional steps

$(7 x - 8) = (9 x + 4)$

Subtract from both sides:

$(7 x - 8) - 9 x = (9 x + 4) - 9 x$

Group like terms:

$(7 x - 9 x) - 8 = (9 x + 4) - 9 x$

Simplify the arithmetic:

$- 2 x - 8 = (9 x + 4) - 9 x$

Group like terms:

$- 2 x - 8 = (9 x - 9 x) + 4$

Simplify the arithmetic:

$- 2 x - 8 = 4$

Add to both sides:

$(- 2 x - 8) + 8 = 4 + 8$

Simplify the arithmetic:

$- 2 x = 4 + 8$

Simplify the arithmetic:

$- 2 x = 12$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{12}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{12}{- 2}$

Simplify the fraction:

$x = \frac{12}{- 2}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 12}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 6 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = - 6$

12 additional steps

$(7 x - 8) = - (9 x + 4)$

Expand the parentheses:

$(7 x - 8) = - 9 x - 4$

Add to both sides:

$(7 x - 8) + 9 x = (- 9 x - 4) + 9 x$

Group like terms:

$(7 x + 9 x) - 8 = (- 9 x - 4) + 9 x$

Simplify the arithmetic:

$16 x - 8 = (- 9 x - 4) + 9 x$

Group like terms:

$16 x - 8 = (- 9 x + 9 x) - 4$

Simplify the arithmetic:

$16 x - 8 = - 4$

Add to both sides:

$(16 x - 8) + 8 = - 4 + 8$

Simplify the arithmetic:

$16 x = - 4 + 8$

Simplify the arithmetic:

$16 x = 4$

Divide both sides by :

$\frac{(16 x)}{16} = \frac{4}{16}$

Simplify the fraction:

$x = \frac{4}{16}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 4)}{(4 \cdot 4)}$

Factor out and cancel the greatest common factor:

$x = \frac{1}{4}$

3. List the solutions

$x = - 6, \frac{1}{4}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 7 x - 8 |$
$y = | 9 x + 4 |$
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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