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Solution - Absolute value equations

Exact form:

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

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

Mixed number form:

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

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

Decimal form:

x = - 1.667, 1

x=-1.667 , 1

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
$| 7 x + 1 | = | x - 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 7 x + 1 \| = \| x - 9 \|$
$x = + y$	$(7 x + 1) = (x - 9)$
$x = - y$	$(7 x + 1) = - (x - 9)$
$+ x = y$	$(7 x + 1) = (x - 9)$
$- x = y$	$- (7 x + 1) = (x - 9)$

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:

$\| x \| = \| y \|$	$\| 7 x + 1 \| = \| x - 9 \|$
$x = + y$ , $+ x = y$	$(7 x + 1) = (x - 9)$
$x = - y$ , $- x = y$	$(7 x + 1) = - (x - 9)$

2. Solve the two equations for x

11 additional steps

$(7 x + 1) = (x - 9)$

Subtract from both sides:

$(7 x + 1) - x = (x - 9) - x$

Group like terms:

$(7 x - x) + 1 = (x - 9) - x$

Simplify the arithmetic:

$6 x + 1 = (x - 9) - x$

Group like terms:

$6 x + 1 = (x - x) - 9$

Simplify the arithmetic:

$6 x + 1 = - 9$

Subtract from both sides:

$(6 x + 1) - 1 = - 9 - 1$

Simplify the arithmetic:

$6 x = - 9 - 1$

Simplify the arithmetic:

$6 x = - 10$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{- 10}{6}$

Simplify the fraction:

$x = \frac{- 10}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 5 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 5}{3}$

11 additional steps

$(7 x + 1) = - (x - 9)$

Expand the parentheses:

$(7 x + 1) = - x + 9$

Add to both sides:

$(7 x + 1) + x = (- x + 9) + x$

Group like terms:

$(7 x + x) + 1 = (- x + 9) + x$

Simplify the arithmetic:

$8 x + 1 = (- x + 9) + x$

Group like terms:

$8 x + 1 = (- x + x) + 9$

Simplify the arithmetic:

$8 x + 1 = 9$

Subtract from both sides:

$(8 x + 1) - 1 = 9 - 1$

Simplify the arithmetic:

$8 x = 9 - 1$

Simplify the arithmetic:

$8 x = 8$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{8}{8}$

Simplify the fraction:

$x = \frac{8}{8}$

Simplify the fraction:

$x = 1$

3. List the solutions

$x = - \frac{5}{3}, 1$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 7 x + 1 |$
$y = | x - 9 |$
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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