Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

a = 2

a=2

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
$| - 3 a + 5 | = | - 3 a + 7 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 3 a + 5 \| = \| - 3 a + 7 \|$
$x = + y$	$(- 3 a + 5) = (- 3 a + 7)$
$x = - y$	$(- 3 a + 5) = - (- 3 a + 7)$
$+ x = y$	$(- 3 a + 5) = (- 3 a + 7)$
$- x = y$	$- (- 3 a + 5) = (- 3 a + 7)$

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:

$\| x \| = \| y \|$	$\| - 3 a + 5 \| = \| - 3 a + 7 \|$
$x = + y$ , $+ x = y$	$(- 3 a + 5) = (- 3 a + 7)$
$x = - y$ , $- x = y$	$(- 3 a + 5) = - (- 3 a + 7)$

2. Solve the two equations for a

5 additional steps

$(- 3 a + 5) = (- 3 a + 7)$

Add to both sides:

$(- 3 a + 5) + 3 a = (- 3 a + 7) + 3 a$

Group like terms:

$(- 3 a + 3 a) + 5 = (- 3 a + 7) + 3 a$

Simplify the arithmetic:

$5 = (- 3 a + 7) + 3 a$

Group like terms:

$5 = (- 3 a + 3 a) + 7$

Simplify the arithmetic:

$5 = 7$

The statement is false:

$5 = 7$

The equation is false so it has no solution.

14 additional steps

$(- 3 a + 5) = - (- 3 a + 7)$

Expand the parentheses:

$(- 3 a + 5) = 3 a - 7$

Subtract from both sides:

$(- 3 a + 5) - 3 a = (3 a - 7) - 3 a$

Group like terms:

$(- 3 a - 3 a) + 5 = (3 a - 7) - 3 a$

Simplify the arithmetic:

$- 6 a + 5 = (3 a - 7) - 3 a$

Group like terms:

$- 6 a + 5 = (3 a - 3 a) - 7$

Simplify the arithmetic:

$- 6 a + 5 = - 7$

Subtract from both sides:

$(- 6 a + 5) - 5 = - 7 - 5$

Simplify the arithmetic:

$- 6 a = - 7 - 5$

Simplify the arithmetic:

$- 6 a = - 12$

Divide both sides by :

$\frac{(- 6 a)}{- 6} = \frac{- 12}{- 6}$

Cancel out the negatives:

$\frac{6 a}{6} = \frac{- 12}{- 6}$

Simplify the fraction:

$a = \frac{- 12}{- 6}$

Cancel out the negatives:

$a = \frac{12}{6}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(2 \cdot 6)}{(1 \cdot 6)}$

Factor out and cancel the greatest common factor:

$a = 2$

3. Graph

Each line represents the function of one side of the equation:
$y = | - 3 a + 5 |$
$y = | - 3 a + 7 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved