Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

a = \frac{3}{4}

a=\frac{3}{4}

Decimal form:

a = 0.75

a=0.75

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
$| - 2 a + 3 | = 2 | a |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for a

10 additional steps

$(- 2 a + 3) = 2 a$

Subtract from both sides:

$(- 2 a + 3) - 2 a = (2 a) - 2 a$

Group like terms:

$(- 2 a - 2 a) + 3 = (2 a) - 2 a$

Simplify the arithmetic:

$- 4 a + 3 = (2 a) - 2 a$

Simplify the arithmetic:

$- 4 a + 3 = 0$

Subtract from both sides:

$(- 4 a + 3) - 3 = 0 - 3$

Simplify the arithmetic:

$- 4 a = 0 - 3$

Simplify the arithmetic:

$- 4 a = - 3$

Divide both sides by :

$\frac{(- 4 a)}{- 4} = \frac{- 3}{- 4}$

Cancel out the negatives:

$\frac{4 a}{4} = \frac{- 3}{- 4}$

Simplify the fraction:

$a = \frac{- 3}{- 4}$

Cancel out the negatives:

$a = \frac{3}{4}$

6 additional steps

$(- 2 a + 3) = 2 \cdot - a$

Group like terms:

$(- 2 a + 3) = (2 \cdot - 1) a$

Multiply the coefficients:

$(- 2 a + 3) = - 2 a$

Add to both sides:

$(- 2 a + 3) + 2 a = (- 2 a) + 2 a$

Group like terms:

$(- 2 a + 2 a) + 3 = (- 2 a) + 2 a$

Simplify the arithmetic:

$3 = (- 2 a) + 2 a$

Simplify the arithmetic:

$3 = 0$

The statement is false:

$3 = 0$

The equation is false so it has no solution.

3. List the solutions

$a = \frac{3}{4}$
(1 solution(s))

4. Graph

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

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 a

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved