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Solution - Absolute value equations

Exact form:

x = \frac{7}{3}, - \frac{1}{2}

x=\frac{7}{3} , -\frac{1}{2}

Mixed number form:

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

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

Decimal form:

x = 2.333, - 0.5

x=2.333 , -0.5

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
$| 9 x - 4 | = | 3 x + 10 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x - 4 \| = \| 3 x + 10 \|$
$x = + y$	$(9 x - 4) = (3 x + 10)$
$x = - y$	$(9 x - 4) = - (3 x + 10)$
$+ x = y$	$(9 x - 4) = (3 x + 10)$
$- x = y$	$- (9 x - 4) = (3 x + 10)$

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:

$\| x \| = \| y \|$	$\| 9 x - 4 \| = \| 3 x + 10 \|$
$x = + y$ , $+ x = y$	$(9 x - 4) = (3 x + 10)$
$x = - y$ , $- x = y$	$(9 x - 4) = - (3 x + 10)$

2. Solve the two equations for x

11 additional steps

$(9 x - 4) = (3 x + 10)$

Subtract from both sides:

$(9 x - 4) - 3 x = (3 x + 10) - 3 x$

Group like terms:

$(9 x - 3 x) - 4 = (3 x + 10) - 3 x$

Simplify the arithmetic:

$6 x - 4 = (3 x + 10) - 3 x$

Group like terms:

$6 x - 4 = (3 x - 3 x) + 10$

Simplify the arithmetic:

$6 x - 4 = 10$

Add to both sides:

$(6 x - 4) + 4 = 10 + 4$

Simplify the arithmetic:

$6 x = 10 + 4$

Simplify the arithmetic:

$6 x = 14$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{14}{6}$

Simplify the fraction:

$x = \frac{14}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(7 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

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

12 additional steps

$(9 x - 4) = - (3 x + 10)$

Expand the parentheses:

$(9 x - 4) = - 3 x - 10$

Add to both sides:

$(9 x - 4) + 3 x = (- 3 x - 10) + 3 x$

Group like terms:

$(9 x + 3 x) - 4 = (- 3 x - 10) + 3 x$

Simplify the arithmetic:

$12 x - 4 = (- 3 x - 10) + 3 x$

Group like terms:

$12 x - 4 = (- 3 x + 3 x) - 10$

Simplify the arithmetic:

$12 x - 4 = - 10$

Add to both sides:

$(12 x - 4) + 4 = - 10 + 4$

Simplify the arithmetic:

$12 x = - 10 + 4$

Simplify the arithmetic:

$12 x = - 6$

Divide both sides by :

$\frac{(12 x)}{12} = \frac{- 6}{12}$

Simplify the fraction:

$x = \frac{- 6}{12}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 6)}{(2 \cdot 6)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{2}$

3. List the solutions

$x = \frac{7}{3}, - \frac{1}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 x - 4 |$
$y = | 3 x + 10 |$
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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