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Solution - Absolute value equations

Exact form:

x = - \frac{9}{2}, \frac{5}{14}

x=-\frac{9}{2} , \frac{5}{14}

Mixed number form:

x = - 4 \frac{1}{2}, \frac{5}{14}

x=-4\frac{1}{2} , \frac{5}{14}

Decimal form:

x = - 4.5, 0.357

x=-4.5 , 0.357

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
$| 8 x + 2 | = | 6 x - 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 x + 2 \| = \| 6 x - 7 \|$
$x = + y$	$(8 x + 2) = (6 x - 7)$
$x = - y$	$(8 x + 2) = - (6 x - 7)$
$+ x = y$	$(8 x + 2) = (6 x - 7)$
$- x = y$	$- (8 x + 2) = (6 x - 7)$

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:

$\| x \| = \| y \|$	$\| 8 x + 2 \| = \| 6 x - 7 \|$
$x = + y$ , $+ x = y$	$(8 x + 2) = (6 x - 7)$
$x = - y$ , $- x = y$	$(8 x + 2) = - (6 x - 7)$

2. Solve the two equations for x

9 additional steps

$(8 x + 2) = (6 x - 7)$

Subtract from both sides:

$(8 x + 2) - 6 x = (6 x - 7) - 6 x$

Group like terms:

$(8 x - 6 x) + 2 = (6 x - 7) - 6 x$

Simplify the arithmetic:

$2 x + 2 = (6 x - 7) - 6 x$

Group like terms:

$2 x + 2 = (6 x - 6 x) - 7$

Simplify the arithmetic:

$2 x + 2 = - 7$

Subtract from both sides:

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

Simplify the arithmetic:

$2 x = - 7 - 2$

Simplify the arithmetic:

$2 x = - 9$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 9}{2}$

Simplify the fraction:

$x = \frac{- 9}{2}$

10 additional steps

$(8 x + 2) = - (6 x - 7)$

Expand the parentheses:

$(8 x + 2) = - 6 x + 7$

Add to both sides:

$(8 x + 2) + 6 x = (- 6 x + 7) + 6 x$

Group like terms:

$(8 x + 6 x) + 2 = (- 6 x + 7) + 6 x$

Simplify the arithmetic:

$14 x + 2 = (- 6 x + 7) + 6 x$

Group like terms:

$14 x + 2 = (- 6 x + 6 x) + 7$

Simplify the arithmetic:

$14 x + 2 = 7$

Subtract from both sides:

$(14 x + 2) - 2 = 7 - 2$

Simplify the arithmetic:

$14 x = 7 - 2$

Simplify the arithmetic:

$14 x = 5$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{5}{14}$

Simplify the fraction:

$x = \frac{5}{14}$

3. List the solutions

$x = - \frac{9}{2}, \frac{5}{14}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 8 x + 2 |$
$y = | 6 x - 7 |$
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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