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Solution - Absolute value equations

Exact form:

x = \frac{13}{10}, - \frac{1}{7}

x=\frac{13}{10} , -\frac{1}{7}

Mixed number form:

x = 1 \frac{3}{10}, - \frac{1}{7}

x=1\frac{3}{10} , -\frac{1}{7}

Decimal form:

x = 1.3, - 0.143

x=1.3 , -0.143

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
$| 17 x - 12 | = | - 3 x + 14 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 17 x - 12 \| = \| - 3 x + 14 \|$
$x = + y$	$(17 x - 12) = (- 3 x + 14)$
$x = - y$	$(17 x - 12) = - (- 3 x + 14)$
$+ x = y$	$(17 x - 12) = (- 3 x + 14)$
$- x = y$	$- (17 x - 12) = (- 3 x + 14)$

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:

$\| x \| = \| y \|$	$\| 17 x - 12 \| = \| - 3 x + 14 \|$
$x = + y$ , $+ x = y$	$(17 x - 12) = (- 3 x + 14)$
$x = - y$ , $- x = y$	$(17 x - 12) = - (- 3 x + 14)$

2. Solve the two equations for x

11 additional steps

$(17 x - 12) = (- 3 x + 14)$

Add to both sides:

$(17 x - 12) + 3 x = (- 3 x + 14) + 3 x$

Group like terms:

$(17 x + 3 x) - 12 = (- 3 x + 14) + 3 x$

Simplify the arithmetic:

$20 x - 12 = (- 3 x + 14) + 3 x$

Group like terms:

$20 x - 12 = (- 3 x + 3 x) + 14$

Simplify the arithmetic:

$20 x - 12 = 14$

Add to both sides:

$(20 x - 12) + 12 = 14 + 12$

Simplify the arithmetic:

$20 x = 14 + 12$

Simplify the arithmetic:

$20 x = 26$

Divide both sides by :

$\frac{(20 x)}{20} = \frac{26}{20}$

Simplify the fraction:

$x = \frac{26}{20}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(13 \cdot 2)}{(10 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{13}{10}$

12 additional steps

$(17 x - 12) = - (- 3 x + 14)$

Expand the parentheses:

$(17 x - 12) = 3 x - 14$

Subtract from both sides:

$(17 x - 12) - 3 x = (3 x - 14) - 3 x$

Group like terms:

$(17 x - 3 x) - 12 = (3 x - 14) - 3 x$

Simplify the arithmetic:

$14 x - 12 = (3 x - 14) - 3 x$

Group like terms:

$14 x - 12 = (3 x - 3 x) - 14$

Simplify the arithmetic:

$14 x - 12 = - 14$

Add to both sides:

$(14 x - 12) + 12 = - 14 + 12$

Simplify the arithmetic:

$14 x = - 14 + 12$

Simplify the arithmetic:

$14 x = - 2$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{- 2}{14}$

Simplify the fraction:

$x = \frac{- 2}{14}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 2)}{(7 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{7}$

3. List the solutions

$x = \frac{13}{10}, - \frac{1}{7}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 17 x - 12 |$
$y = | - 3 x + 14 |$
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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