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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = - 2, 0.5

x=-2 , 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
$| 3 x + 1 | = | x - 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x + 1 \| = \| x - 3 \|$
$x = + y$	$(3 x + 1) = (x - 3)$
$x = - y$	$(3 x + 1) = - (x - 3)$
$+ x = y$	$(3 x + 1) = (x - 3)$
$- x = y$	$- (3 x + 1) = (x - 3)$

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:

$\| x \| = \| y \|$	$\| 3 x + 1 \| = \| x - 3 \|$
$x = + y$ , $+ x = y$	$(3 x + 1) = (x - 3)$
$x = - y$ , $- x = y$	$(3 x + 1) = - (x - 3)$

2. Solve the two equations for x

11 additional steps

$(3 x + 1) = (x - 3)$

Subtract from both sides:

$(3 x + 1) - x = (x - 3) - x$

Group like terms:

$(3 x - x) + 1 = (x - 3) - x$

Simplify the arithmetic:

$2 x + 1 = (x - 3) - x$

Group like terms:

$2 x + 1 = (x - x) - 3$

Simplify the arithmetic:

$2 x + 1 = - 3$

Subtract from both sides:

$(2 x + 1) - 1 = - 3 - 1$

Simplify the arithmetic:

$2 x = - 3 - 1$

Simplify the arithmetic:

$2 x = - 4$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{- 4}{2}$

Simplify the fraction:

$x = \frac{- 4}{2}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = - 2$

12 additional steps

$(3 x + 1) = - (x - 3)$

Expand the parentheses:

$(3 x + 1) = - x + 3$

Add to both sides:

$(3 x + 1) + x = (- x + 3) + x$

Group like terms:

$(3 x + x) + 1 = (- x + 3) + x$

Simplify the arithmetic:

$4 x + 1 = (- x + 3) + x$

Group like terms:

$4 x + 1 = (- x + x) + 3$

Simplify the arithmetic:

$4 x + 1 = 3$

Subtract from both sides:

$(4 x + 1) - 1 = 3 - 1$

Simplify the arithmetic:

$4 x = 3 - 1$

Simplify the arithmetic:

$4 x = 2$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{2}{4}$

Simplify the fraction:

$x = \frac{2}{4}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

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

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 x + 1 |$
$y = | x - 3 |$
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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