Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

n = 10, 0

n=10 , 0

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
$| - 4 n + 5 | = | - 3 n - 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 4 n + 5 \| = \| - 3 n - 5 \|$
$x = + y$	$(- 4 n + 5) = (- 3 n - 5)$
$x = - y$	$(- 4 n + 5) = - (- 3 n - 5)$
$+ x = y$	$(- 4 n + 5) = (- 3 n - 5)$
$- x = y$	$- (- 4 n + 5) = (- 3 n - 5)$

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:

$\| x \| = \| y \|$	$\| - 4 n + 5 \| = \| - 3 n - 5 \|$
$x = + y$ , $+ x = y$	$(- 4 n + 5) = (- 3 n - 5)$
$x = - y$ , $- x = y$	$(- 4 n + 5) = - (- 3 n - 5)$

2. Solve the two equations for n

10 additional steps

$(- 4 n + 5) = (- 3 n - 5)$

Add to both sides:

$(- 4 n + 5) + 3 n = (- 3 n - 5) + 3 n$

Group like terms:

$(- 4 n + 3 n) + 5 = (- 3 n - 5) + 3 n$

Simplify the arithmetic:

$- n + 5 = (- 3 n - 5) + 3 n$

Group like terms:

$- n + 5 = (- 3 n + 3 n) - 5$

Simplify the arithmetic:

$- n + 5 = - 5$

Subtract from both sides:

$(- n + 5) - 5 = - 5 - 5$

Simplify the arithmetic:

$- n = - 5 - 5$

Simplify the arithmetic:

$- n = - 10$

Multiply both sides by :

$- n \cdot - 1 = - 10 \cdot - 1$

Remove the one(s):

$n = - 10 \cdot - 1$

Simplify the arithmetic:

$n = 10$

9 additional steps

$(- 4 n + 5) = - (- 3 n - 5)$

Expand the parentheses:

$(- 4 n + 5) = 3 n + 5$

Subtract from both sides:

$(- 4 n + 5) - 3 n = (3 n + 5) - 3 n$

Group like terms:

$(- 4 n - 3 n) + 5 = (3 n + 5) - 3 n$

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$- 7 n + 5 = 5$

Subtract from both sides:

$(- 7 n + 5) - 5 = 5 - 5$

Simplify the arithmetic:

$- 7 n = 5 - 5$

Simplify the arithmetic:

$- 7 n = 0$

Divide both sides by the coefficient:

$n = 0$

3. List the solutions

$n = 10, 0$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 4 n + 5 |$
$y = | - 3 n - 5 |$
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 n

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 n

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved