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Solution - Absolute value equations

Exact form:

x = 4, 11

x=4 , 11

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
$| 2 x - 1 | = | - 4 x + 23 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = \| - 4 x + 23 \|$
$x = + y$	$(2 x - 1) = (- 4 x + 23)$
$x = - y$	$(2 x - 1) = - (- 4 x + 23)$
$+ x = y$	$(2 x - 1) = (- 4 x + 23)$
$- x = y$	$- (2 x - 1) = (- 4 x + 23)$

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:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = \| - 4 x + 23 \|$
$x = + y$ , $+ x = y$	$(2 x - 1) = (- 4 x + 23)$
$x = - y$ , $- x = y$	$(2 x - 1) = - (- 4 x + 23)$

2. Solve the two equations for x

11 additional steps

$(2 x - 1) = (- 4 x + 23)$

Add to both sides:

$(2 x - 1) + 4 x = (- 4 x + 23) + 4 x$

Group like terms:

$(2 x + 4 x) - 1 = (- 4 x + 23) + 4 x$

Simplify the arithmetic:

$6 x - 1 = (- 4 x + 23) + 4 x$

Group like terms:

$6 x - 1 = (- 4 x + 4 x) + 23$

Simplify the arithmetic:

$6 x - 1 = 23$

Add to both sides:

$(6 x - 1) + 1 = 23 + 1$

Simplify the arithmetic:

$6 x = 23 + 1$

Simplify the arithmetic:

$6 x = 24$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{24}{6}$

Simplify the fraction:

$x = \frac{24}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(4 \cdot 6)}{(1 \cdot 6)}$

Factor out and cancel the greatest common factor:

$x = 4$

14 additional steps

$(2 x - 1) = - (- 4 x + 23)$

Expand the parentheses:

$(2 x - 1) = 4 x - 23$

Subtract from both sides:

$(2 x - 1) - 4 x = (4 x - 23) - 4 x$

Group like terms:

$(2 x - 4 x) - 1 = (4 x - 23) - 4 x$

Simplify the arithmetic:

$- 2 x - 1 = (4 x - 23) - 4 x$

Group like terms:

$- 2 x - 1 = (4 x - 4 x) - 23$

Simplify the arithmetic:

$- 2 x - 1 = - 23$

Add to both sides:

$(- 2 x - 1) + 1 = - 23 + 1$

Simplify the arithmetic:

$- 2 x = - 23 + 1$

Simplify the arithmetic:

$- 2 x = - 22$

Divide both sides by :

$\frac{(- 2 x)}{- 2} = \frac{- 22}{- 2}$

Cancel out the negatives:

$\frac{2 x}{2} = \frac{- 22}{- 2}$

Simplify the fraction:

$x = \frac{- 22}{- 2}$

Cancel out the negatives:

$x = \frac{22}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(11 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 11$

3. List the solutions

$x = 4, 11$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 1 |$
$y = | - 4 x + 23 |$
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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