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Solution - Absolute value equations

Exact form:

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

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

Decimal form:

x = 2, - 0.333

x=2 , -0.333

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
$| 2 x + 3 | = | 4 x - 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x + 3 \| = \| 4 x - 1 \|$
$x = + y$	$(2 x + 3) = (4 x - 1)$
$x = - y$	$(2 x + 3) = - (4 x - 1)$
$+ x = y$	$(2 x + 3) = (4 x - 1)$
$- x = y$	$- (2 x + 3) = (4 x - 1)$

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:

$\| x \| = \| y \|$	$\| 2 x + 3 \| = \| 4 x - 1 \|$
$x = + y$ , $+ x = y$	$(2 x + 3) = (4 x - 1)$
$x = - y$ , $- x = y$	$(2 x + 3) = - (4 x - 1)$

2. Solve the two equations for x

13 additional steps

$(2 x + 3) = (4 x - 1)$

Subtract from both sides:

$(2 x + 3) - 4 x = (4 x - 1) - 4 x$

Group like terms:

$(2 x - 4 x) + 3 = (4 x - 1) - 4 x$

Simplify the arithmetic:

$- 2 x + 3 = (4 x - 1) - 4 x$

Group like terms:

$- 2 x + 3 = (4 x - 4 x) - 1$

Simplify the arithmetic:

$- 2 x + 3 = - 1$

Subtract from both sides:

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

Simplify the arithmetic:

$- 2 x = - 1 - 3$

Simplify the arithmetic:

$- 2 x = - 4$

Divide both sides by :

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

Cancel out the negatives:

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

Simplify the fraction:

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

Cancel out the negatives:

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

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

$(2 x + 3) = - (4 x - 1)$

Expand the parentheses:

$(2 x + 3) = - 4 x + 1$

Add to both sides:

$(2 x + 3) + 4 x = (- 4 x + 1) + 4 x$

Group like terms:

$(2 x + 4 x) + 3 = (- 4 x + 1) + 4 x$

Simplify the arithmetic:

$6 x + 3 = (- 4 x + 1) + 4 x$

Group like terms:

$6 x + 3 = (- 4 x + 4 x) + 1$

Simplify the arithmetic:

$6 x + 3 = 1$

Subtract from both sides:

$(6 x + 3) - 3 = 1 - 3$

Simplify the arithmetic:

$6 x = 1 - 3$

Simplify the arithmetic:

$6 x = - 2$

Divide both sides by :

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

Simplify the fraction:

$x = \frac{- 2}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(- 1 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{3}$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x + 3 |$
$y = | 4 x - 1 |$
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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