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Solution - Absolute value equations

Exact form:

x = 1, - 5

x=1 , -5

See steps

Step-by-step explanation

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

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

Add $- | 3 x |$ to both sides of the equation:

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

Simplify the arithmetic

$| 2 x - 5 | = - | 3 x |$

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
$| 2 x - 5 | = - | 3 x |$
without the absolute value bars:

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

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:

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

3. Solve the two equations for x

8 additional steps

$(2 x - 5) = - 3 x$

Add to both sides:

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

Simplify the arithmetic:

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

Add to both sides:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$5 x = 5$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{5}{5}$

Simplify the fraction:

$x = \frac{5}{5}$

Simplify the fraction:

$x = 1$

11 additional steps

$(2 x - 5) = - - 3 x$

Group like terms:

$(2 x - 5) = (- 1 \cdot - 3) x$

Multiply the coefficients:

$(2 x - 5) = 3 x$

Subtract from both sides:

$(2 x - 5) - 3 x = (3 x) - 3 x$

Group like terms:

$(2 x - 3 x) - 5 = (3 x) - 3 x$

Simplify the arithmetic:

$- x - 5 = (3 x) - 3 x$

Simplify the arithmetic:

$- x - 5 = 0$

Add to both sides:

$(- x - 5) + 5 = 0 + 5$

Simplify the arithmetic:

$- x = 0 + 5$

Simplify the arithmetic:

$- x = 5$

Multiply both sides by :

$- x \cdot - 1 = 5 \cdot - 1$

Remove the one(s):

$x = 5 \cdot - 1$

Simplify the arithmetic:

$x = - 5$

4. List the solutions

$x = 1, - 5$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 2 x - 5 |$
$y = - | 3 x |$
The equation is true where the two lines cross.

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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 x

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 x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved