Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = - \frac{1}{41}, \frac{1}{45}

x=-\frac{1}{41} , \frac{1}{45}

Decimal form:

x = - 0.024, 0.022

x=-0.024 , 0.022

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
$| 2 x - 1 | = | 43 x |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = \| 43 x \|$
$x = + y$	$(2 x - 1) = (43 x)$
$x = - y$	$(2 x - 1) = - (43 x)$
$+ x = y$	$(2 x - 1) = (43 x)$
$- x = y$	$- (2 x - 1) = (43 x)$

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:

$\| x \| = \| y \|$	$\| 2 x - 1 \| = \| 43 x \|$
$x = + y$ , $+ x = y$	$(2 x - 1) = (43 x)$
$x = - y$ , $- x = y$	$(2 x - 1) = - (43 x)$

2. Solve the two equations for x

10 additional steps

$(2 x - 1) = 43 x$

Subtract from both sides:

$(2 x - 1) - 43 x = (43 x) - 43 x$

Group like terms:

$(2 x - 43 x) - 1 = (43 x) - 43 x$

Simplify the arithmetic:

$- 41 x - 1 = (43 x) - 43 x$

Simplify the arithmetic:

$- 41 x - 1 = 0$

Add to both sides:

$(- 41 x - 1) + 1 = 0 + 1$

Simplify the arithmetic:

$- 41 x = 0 + 1$

Simplify the arithmetic:

$- 41 x = 1$

Divide both sides by :

$\frac{(- 41 x)}{- 41} = \frac{1}{- 41}$

Cancel out the negatives:

$\frac{41 x}{41} = \frac{1}{- 41}$

Simplify the fraction:

$x = \frac{1}{- 41}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 1}{41}$

7 additional steps

$(2 x - 1) = - 43 x$

Add to both sides:

$(2 x - 1) + 1 = (- 43 x) + 1$

Simplify the arithmetic:

$2 x = (- 43 x) + 1$

Add to both sides:

$(2 x) + 43 x = ((- 43 x) + 1) + 43 x$

Simplify the arithmetic:

$45 x = ((- 43 x) + 1) + 43 x$

Group like terms:

$45 x = (- 43 x + 43 x) + 1$

Simplify the arithmetic:

$45 x = 1$

Divide both sides by :

$\frac{(45 x)}{45} = \frac{1}{45}$

Simplify the fraction:

$x = \frac{1}{45}$

3. List the solutions

$x = - \frac{1}{41}, \frac{1}{45}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 1 |$
$y = | 43 x |$
The equation is true where the two lines cross.

How did we do?

Please leave us feedback.

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

Latest Related Drills Solved