Enter an equation or problem

Camera input is not recognized!

Solution - Absolute value equations

Exact form:

x = \frac{1}{7}, 29

x=\frac{1}{7} , 29

Decimal form:

x = 0.143, 29

x=0.143 , 29

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
$| - 4 x + 15 | = | 3 x + 14 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| - 4 x + 15 \| = \| 3 x + 14 \|$
$x = + y$	$(- 4 x + 15) = (3 x + 14)$
$x = - y$	$(- 4 x + 15) = - (3 x + 14)$
$+ x = y$	$(- 4 x + 15) = (3 x + 14)$
$- x = y$	$- (- 4 x + 15) = (3 x + 14)$

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:

$\| x \| = \| y \|$	$\| - 4 x + 15 \| = \| 3 x + 14 \|$
$x = + y$ , $+ x = y$	$(- 4 x + 15) = (3 x + 14)$
$x = - y$ , $- x = y$	$(- 4 x + 15) = - (3 x + 14)$

2. Solve the two equations for x

11 additional steps

$(- 4 x + 15) = (3 x + 14)$

Subtract from both sides:

$(- 4 x + 15) - 3 x = (3 x + 14) - 3 x$

Group like terms:

$(- 4 x - 3 x) + 15 = (3 x + 14) - 3 x$

Simplify the arithmetic:

$- 7 x + 15 = (3 x + 14) - 3 x$

Group like terms:

$- 7 x + 15 = (3 x - 3 x) + 14$

Simplify the arithmetic:

$- 7 x + 15 = 14$

Subtract from both sides:

$(- 7 x + 15) - 15 = 14 - 15$

Simplify the arithmetic:

$- 7 x = 14 - 15$

Simplify the arithmetic:

$- 7 x = - 1$

Divide both sides by :

$\frac{(- 7 x)}{- 7} = \frac{- 1}{- 7}$

Cancel out the negatives:

$\frac{7 x}{7} = \frac{- 1}{- 7}$

Simplify the fraction:

$x = \frac{- 1}{- 7}$

Cancel out the negatives:

$x = \frac{1}{7}$

11 additional steps

$(- 4 x + 15) = - (3 x + 14)$

Expand the parentheses:

$(- 4 x + 15) = - 3 x - 14$

Add to both sides:

$(- 4 x + 15) + 3 x = (- 3 x - 14) + 3 x$

Group like terms:

$(- 4 x + 3 x) + 15 = (- 3 x - 14) + 3 x$

Simplify the arithmetic:

$- x + 15 = (- 3 x - 14) + 3 x$

Group like terms:

$- x + 15 = (- 3 x + 3 x) - 14$

Simplify the arithmetic:

$- x + 15 = - 14$

Subtract from both sides:

$(- x + 15) - 15 = - 14 - 15$

Simplify the arithmetic:

$- x = - 14 - 15$

Simplify the arithmetic:

$- x = - 29$

Multiply both sides by :

$- x \cdot - 1 = - 29 \cdot - 1$

Remove the one(s):

$x = - 29 \cdot - 1$

Simplify the arithmetic:

$x = 29$

3. List the solutions

$x = \frac{1}{7}, 29$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | - 4 x + 15 |$
$y = | 3 x + 14 |$
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