Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{5}{4}, \frac{3}{14}

x=\frac{5}{4} , \frac{3}{14}

Mixed number form:

x = 1 \frac{1}{4}, \frac{3}{14}

x=1\frac{1}{4} , \frac{3}{14}

Decimal form:

x = 1.25, 0.214

x=1.25 , 0.214

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
$| 9 x - 4 | = | 5 x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x - 4 \| = \| 5 x + 1 \|$
$x = + y$	$(9 x - 4) = (5 x + 1)$
$x = - y$	$(9 x - 4) = - (5 x + 1)$
$+ x = y$	$(9 x - 4) = (5 x + 1)$
$- x = y$	$- (9 x - 4) = (5 x + 1)$

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:

$\| x \| = \| y \|$	$\| 9 x - 4 \| = \| 5 x + 1 \|$
$x = + y$ , $+ x = y$	$(9 x - 4) = (5 x + 1)$
$x = - y$ , $- x = y$	$(9 x - 4) = - (5 x + 1)$

2. Solve the two equations for x

9 additional steps

$(9 x - 4) = (5 x + 1)$

Subtract from both sides:

$(9 x - 4) - 5 x = (5 x + 1) - 5 x$

Group like terms:

$(9 x - 5 x) - 4 = (5 x + 1) - 5 x$

Simplify the arithmetic:

$4 x - 4 = (5 x + 1) - 5 x$

Group like terms:

$4 x - 4 = (5 x - 5 x) + 1$

Simplify the arithmetic:

$4 x - 4 = 1$

Add to both sides:

$(4 x - 4) + 4 = 1 + 4$

Simplify the arithmetic:

$4 x = 1 + 4$

Simplify the arithmetic:

$4 x = 5$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{5}{4}$

Simplify the fraction:

$x = \frac{5}{4}$

10 additional steps

$(9 x - 4) = - (5 x + 1)$

Expand the parentheses:

$(9 x - 4) = - 5 x - 1$

Add to both sides:

$(9 x - 4) + 5 x = (- 5 x - 1) + 5 x$

Group like terms:

$(9 x + 5 x) - 4 = (- 5 x - 1) + 5 x$

Simplify the arithmetic:

$14 x - 4 = (- 5 x - 1) + 5 x$

Group like terms:

$14 x - 4 = (- 5 x + 5 x) - 1$

Simplify the arithmetic:

$14 x - 4 = - 1$

Add to both sides:

$(14 x - 4) + 4 = - 1 + 4$

Simplify the arithmetic:

$14 x = - 1 + 4$

Simplify the arithmetic:

$14 x = 3$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{3}{14}$

Simplify the fraction:

$x = \frac{3}{14}$

3. List the solutions

$x = \frac{5}{4}, \frac{3}{14}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 x - 4 |$
$y = | 5 x + 1 |$
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 x

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 x

3. List the solutions

4. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved