Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

m = - 7, \frac{3}{5}

m=-7 , \frac{3}{5}

Decimal form:

m = - 7, 0.6

m=-7 , 0.6

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 3 m + 2 | + | - 2 m + 5 | = 0$

Add $- | - 2 m + 5 |$ to both sides of the equation:

$| 3 m + 2 | + | - 2 m + 5 | - | - 2 m + 5 | = - | - 2 m + 5 |$

Simplify the arithmetic

$| 3 m + 2 | = - | - 2 m + 5 |$

2. 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
$| 3 m + 2 | = - | - 2 m + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 m + 2 \| = - \| - 2 m + 5 \|$
$x = + y$	$(3 m + 2) = - (- 2 m + 5)$
$x = - y$	$(3 m + 2) = - - (- 2 m + 5)$
$+ x = y$	$(3 m + 2) = - (- 2 m + 5)$
$- x = y$	$- (3 m + 2) = - (- 2 m + 5)$

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:

$\| x \| = \| y \|$	$\| 3 m + 2 \| = - \| - 2 m + 5 \|$
$x = + y$ , $+ x = y$	$(3 m + 2) = - (- 2 m + 5)$
$x = - y$ , $- x = y$	$(3 m + 2) = - - (- 2 m + 5)$

3. Solve the two equations for m

8 additional steps

$(3 m + 2) = - (- 2 m + 5)$

Expand the parentheses:

$(3 m + 2) = 2 m - 5$

Subtract from both sides:

$(3 m + 2) - 2 m = (2 m - 5) - 2 m$

Group like terms:

$(3 m - 2 m) + 2 = (2 m - 5) - 2 m$

Simplify the arithmetic:

$m + 2 = (2 m - 5) - 2 m$

Group like terms:

$m + 2 = (2 m - 2 m) - 5$

Simplify the arithmetic:

$m + 2 = - 5$

Subtract from both sides:

$(m + 2) - 2 = - 5 - 2$

Simplify the arithmetic:

$m = - 5 - 2$

Simplify the arithmetic:

$m = - 7$

10 additional steps

$(3 m + 2) = - (- (- 2 m + 5))$

NT_MSLUS_MAINSTEP_RESOLVE_DOUBLE_MINUS:

$(3 m + 2) = - 2 m + 5$

Add to both sides:

$(3 m + 2) + 2 m = (- 2 m + 5) + 2 m$

Group like terms:

$(3 m + 2 m) + 2 = (- 2 m + 5) + 2 m$

Simplify the arithmetic:

$5 m + 2 = (- 2 m + 5) + 2 m$

Group like terms:

$5 m + 2 = (- 2 m + 2 m) + 5$

Simplify the arithmetic:

$5 m + 2 = 5$

Subtract from both sides:

$(5 m + 2) - 2 = 5 - 2$

Simplify the arithmetic:

$5 m = 5 - 2$

Simplify the arithmetic:

$5 m = 3$

Divide both sides by :

$\frac{(5 m)}{5} = \frac{3}{5}$

Simplify the fraction:

$m = \frac{3}{5}$

4. List the solutions

$m = - 7, \frac{3}{5}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 3 m + 2 |$
$y = - | - 2 m + 5 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for m

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for m

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved