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Solution - Absolute value equations

Exact form:

x = \frac{3}{25}, - \frac{3}{23}

x=\frac{3}{25} , -\frac{3}{23}

Decimal form:

x = 0.12, - 0.130

x=0.12 , -0.130

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| - x + 3 | - | 24 x | = 0$

Add $| 24 x |$ to both sides of the equation:

$| - x + 3 | - | 24 x | + | 24 x | = | 24 x |$

Simplify the arithmetic

$| - x + 3 | = | 24 x |$

2. 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
$| - x + 3 | = | 24 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - x + 3 \| = \| 24 x \|$
$x = + y$	$(- x + 3) = (24 x)$
$x = - y$	$(- x + 3) = (- (24 x))$
$+ x = y$	$(- x + 3) = (24 x)$
$- x = y$	$- (- x + 3) = (24 x)$

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:

$\| x \| = \| y \|$	$\| - x + 3 \| = \| 24 x \|$
$x = + y$ , $+ x = y$	$(- x + 3) = (24 x)$
$x = - y$ , $- x = y$	$(- x + 3) = (- (24 x))$

3. Solve the two equations for x

10 additional steps

$(- x + 3) = 24 x$

Subtract from both sides:

$(- x + 3) - 24 x = (24 x) - 24 x$

Group like terms:

$(- x - 24 x) + 3 = (24 x) - 24 x$

Simplify the arithmetic:

$- 25 x + 3 = (24 x) - 24 x$

Simplify the arithmetic:

$- 25 x + 3 = 0$

Subtract from both sides:

$(- 25 x + 3) - 3 = 0 - 3$

Simplify the arithmetic:

$- 25 x = 0 - 3$

Simplify the arithmetic:

$- 25 x = - 3$

Divide both sides by :

$\frac{(- 25 x)}{- 25} = \frac{- 3}{- 25}$

Cancel out the negatives:

$\frac{25 x}{25} = \frac{- 3}{- 25}$

Simplify the fraction:

$x = \frac{- 3}{- 25}$

Cancel out the negatives:

$x = \frac{3}{25}$

7 additional steps

$(- x + 3) = - 24 x$

Subtract from both sides:

$(- x + 3) - 3 = (- 24 x) - 3$

Simplify the arithmetic:

$- x = (- 24 x) - 3$

Add to both sides:

$- x + 24 x = ((- 24 x) - 3) + 24 x$

Simplify the arithmetic:

$23 x = ((- 24 x) - 3) + 24 x$

Group like terms:

$23 x = (- 24 x + 24 x) - 3$

Simplify the arithmetic:

$23 x = - 3$

Divide both sides by :

$\frac{(23 x)}{23} = \frac{- 3}{23}$

Simplify the fraction:

$x = \frac{- 3}{23}$

4. List the solutions

$x = \frac{3}{25}, - \frac{3}{23}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | - x + 3 |$
$y = | 24 x |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved