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Solution - Absolute value equations

Exact form:

x = \frac{16}{3}, - 2

x=\frac{16}{3} , -2

Mixed number form:

x = 5 \frac{1}{3}, - 2

x=5\frac{1}{3} , -2

Decimal form:

x = 5.333, - 2

x=5.333 , -2

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
$| - 2 x + 7 | = | x - 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 x + 7 \| = \| x - 9 \|$
$x = + y$	$(- 2 x + 7) = (x - 9)$
$x = - y$	$(- 2 x + 7) = - (x - 9)$
$+ x = y$	$(- 2 x + 7) = (x - 9)$
$- x = y$	$- (- 2 x + 7) = (x - 9)$

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:

$\| x \| = \| y \|$	$\| - 2 x + 7 \| = \| x - 9 \|$
$x = + y$ , $+ x = y$	$(- 2 x + 7) = (x - 9)$
$x = - y$ , $- x = y$	$(- 2 x + 7) = - (x - 9)$

2. Solve the two equations for x

11 additional steps

$(- 2 x + 7) = (x - 9)$

Subtract from both sides:

$(- 2 x + 7) - x = (x - 9) - x$

Group like terms:

$(- 2 x - x) + 7 = (x - 9) - x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 3 x + 7 = - 9$

Subtract from both sides:

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

Simplify the arithmetic:

$- 3 x = - 9 - 7$

Simplify the arithmetic:

$- 3 x = - 16$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{- 16}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{- 16}{- 3}$

Simplify the fraction:

$x = \frac{- 16}{- 3}$

Cancel out the negatives:

$x = \frac{16}{3}$

11 additional steps

$(- 2 x + 7) = - (x - 9)$

Expand the parentheses:

$(- 2 x + 7) = - x + 9$

Add to both sides:

$(- 2 x + 7) + x = (- x + 9) + x$

Group like terms:

$(- 2 x + x) + 7 = (- x + 9) + x$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- x + 7 = 9$

Subtract from both sides:

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

Simplify the arithmetic:

$- x = 9 - 7$

Simplify the arithmetic:

$- x = 2$

Multiply both sides by :

$- x \cdot - 1 = 2 \cdot - 1$

Remove the one(s):

$x = 2 \cdot - 1$

Simplify the arithmetic:

$x = - 2$

3. List the solutions

$x = \frac{16}{3}, - 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 2 x + 7 |$
$y = | x - 9 |$
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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