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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = - 0.429, - 7

x=-0.429 , -7

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
$| 2 x - 9 | = - | 5 x + 12 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 9 \| = - \| 5 x + 12 \|$
$x = + y$	$(2 x - 9) = - (5 x + 12)$
$x = - y$	$(2 x - 9) = - (- (5 x + 12))$
$+ x = y$	$(2 x - 9) = - (5 x + 12)$
$- x = y$	$- (2 x - 9) = - (5 x + 12)$

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:

$\| x \| = \| y \|$	$\| 2 x - 9 \| = - \| 5 x + 12 \|$
$x = + y$ , $+ x = y$	$(2 x - 9) = - (5 x + 12)$
$x = - y$ , $- x = y$	$(2 x - 9) = - (- (5 x + 12))$

2. Solve the two equations for x

10 additional steps

$(2 x - 9) = - (5 x + 12)$

Expand the parentheses:

$(2 x - 9) = - 5 x - 12$

Add to both sides:

$(2 x - 9) + 5 x = (- 5 x - 12) + 5 x$

Group like terms:

$(2 x + 5 x) - 9 = (- 5 x - 12) + 5 x$

Simplify the arithmetic:

$7 x - 9 = (- 5 x - 12) + 5 x$

Group like terms:

$7 x - 9 = (- 5 x + 5 x) - 12$

Simplify the arithmetic:

$7 x - 9 = - 12$

Add to both sides:

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

Simplify the arithmetic:

$7 x = - 12 + 9$

Simplify the arithmetic:

$7 x = - 3$

Divide both sides by :

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

Simplify the fraction:

$x = \frac{- 3}{7}$

14 additional steps

$(2 x - 9) = - (- (5 x + 12))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(2 x - 9) = 5 x + 12$

Subtract from both sides:

$(2 x - 9) - 5 x = (5 x + 12) - 5 x$

Group like terms:

$(2 x - 5 x) - 9 = (5 x + 12) - 5 x$

Simplify the arithmetic:

$- 3 x - 9 = (5 x + 12) - 5 x$

Group like terms:

$- 3 x - 9 = (5 x - 5 x) + 12$

Simplify the arithmetic:

$- 3 x - 9 = 12$

Add to both sides:

$(- 3 x - 9) + 9 = 12 + 9$

Simplify the arithmetic:

$- 3 x = 12 + 9$

Simplify the arithmetic:

$- 3 x = 21$

Divide both sides by :

$\frac{(- 3 x)}{- 3} = \frac{21}{- 3}$

Cancel out the negatives:

$\frac{3 x}{3} = \frac{21}{- 3}$

Simplify the fraction:

$x = \frac{21}{- 3}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 21}{3}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 7 \cdot 3)}{(1 \cdot 3)}$

Factor out and cancel the greatest common factor:

$x = - 7$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 9 |$
$y = - | 5 x + 12 |$
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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