Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

a = \frac{7}{2}

a=\frac{7}{2}

Mixed number form:

a = 3 \frac{1}{2}

a=3\frac{1}{2}

Decimal form:

a = 3.5

a=3.5

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
$| - 2 a + 5 | = | - 2 a + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 2 a + 5 \| = \| - 2 a + 9 \|$
$x = + y$	$(- 2 a + 5) = (- 2 a + 9)$
$x = - y$	$(- 2 a + 5) = - (- 2 a + 9)$
$+ x = y$	$(- 2 a + 5) = (- 2 a + 9)$
$- x = y$	$- (- 2 a + 5) = (- 2 a + 9)$

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:

$\| x \| = \| y \|$	$\| - 2 a + 5 \| = \| - 2 a + 9 \|$
$x = + y$ , $+ x = y$	$(- 2 a + 5) = (- 2 a + 9)$
$x = - y$ , $- x = y$	$(- 2 a + 5) = - (- 2 a + 9)$

2. Solve the two equations for a

5 additional steps

$(- 2 a + 5) = (- 2 a + 9)$

Add to both sides:

$(- 2 a + 5) + 2 a = (- 2 a + 9) + 2 a$

Group like terms:

$(- 2 a + 2 a) + 5 = (- 2 a + 9) + 2 a$

Simplify the arithmetic:

$5 = (- 2 a + 9) + 2 a$

Group like terms:

$5 = (- 2 a + 2 a) + 9$

Simplify the arithmetic:

$5 = 9$

The statement is false:

$5 = 9$

The equation is false so it has no solution.

14 additional steps

$(- 2 a + 5) = - (- 2 a + 9)$

Expand the parentheses:

$(- 2 a + 5) = 2 a - 9$

Subtract from both sides:

$(- 2 a + 5) - 2 a = (2 a - 9) - 2 a$

Group like terms:

$(- 2 a - 2 a) + 5 = (2 a - 9) - 2 a$

Simplify the arithmetic:

$- 4 a + 5 = (2 a - 9) - 2 a$

Group like terms:

$- 4 a + 5 = (2 a - 2 a) - 9$

Simplify the arithmetic:

$- 4 a + 5 = - 9$

Subtract from both sides:

$(- 4 a + 5) - 5 = - 9 - 5$

Simplify the arithmetic:

$- 4 a = - 9 - 5$

Simplify the arithmetic:

$- 4 a = - 14$

Divide both sides by :

$\frac{(- 4 a)}{- 4} = \frac{- 14}{- 4}$

Cancel out the negatives:

$\frac{4 a}{4} = \frac{- 14}{- 4}$

Simplify the fraction:

$a = \frac{- 14}{- 4}$

Cancel out the negatives:

$a = \frac{14}{4}$

Find the greatest common factor of the numerator and denominator:

$a = \frac{(7 \cdot 2)}{(2 \cdot 2)}$

Factor out and cancel the greatest common factor:

$a = \frac{7}{2}$

3. Graph

Each line represents the function of one side of the equation:
$y = | - 2 a + 5 |$
$y = | - 2 a + 9 |$
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