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Solution - Absolute value equations

Exact form:

x = 22, 2

x=22 , 2

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
$| 3 x + 4 | = 2 | 2 x - 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 3 x + 4 \| = 2 \| 2 x - 9 \|$
$x = + y$	$(3 x + 4) = 2 (2 x - 9)$
$x = - y$	$(3 x + 4) = 2 (- (2 x - 9))$
$+ x = y$	$(3 x + 4) = 2 (2 x - 9)$
$- x = y$	$- (3 x + 4) = 2 (2 x - 9)$

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:

$\| x \| = \| y \|$	$\| 3 x + 4 \| = 2 \| 2 x - 9 \|$
$x = + y$ , $+ x = y$	$(3 x + 4) = 2 (2 x - 9)$
$x = - y$ , $- x = y$	$(3 x + 4) = 2 (- (2 x - 9))$

2. Solve the two equations for x

13 additional steps

$(3 x + 4) = 2 \cdot (2 x - 9)$

Expand the parentheses:

$(3 x + 4) = 2 \cdot 2 x + 2 \cdot - 9$

Multiply the coefficients:

$(3 x + 4) = 4 x + 2 \cdot - 9$

Simplify the arithmetic:

$(3 x + 4) = 4 x - 18$

Subtract from both sides:

$(3 x + 4) - 4 x = (4 x - 18) - 4 x$

Group like terms:

$(3 x - 4 x) + 4 = (4 x - 18) - 4 x$

Simplify the arithmetic:

$- x + 4 = (4 x - 18) - 4 x$

Group like terms:

$- x + 4 = (4 x - 4 x) - 18$

Simplify the arithmetic:

$- x + 4 = - 18$

Subtract from both sides:

$(- x + 4) - 4 = - 18 - 4$

Simplify the arithmetic:

$- x = - 18 - 4$

Simplify the arithmetic:

$- x = - 22$

Multiply both sides by :

$- x \cdot - 1 = - 22 \cdot - 1$

Remove the one(s):

$x = - 22 \cdot - 1$

Simplify the arithmetic:

$x = 22$

15 additional steps

$(3 x + 4) = 2 \cdot (- (2 x - 9))$

Expand the parentheses:

$(3 x + 4) = 2 \cdot (- 2 x + 9)$

Expand the parentheses:

$(3 x + 4) = 2 \cdot - 2 x + 2 \cdot 9$

Multiply the coefficients:

$(3 x + 4) = - 4 x + 2 \cdot 9$

Simplify the arithmetic:

$(3 x + 4) = - 4 x + 18$

Add to both sides:

$(3 x + 4) + 4 x = (- 4 x + 18) + 4 x$

Group like terms:

$(3 x + 4 x) + 4 = (- 4 x + 18) + 4 x$

Simplify the arithmetic:

$7 x + 4 = (- 4 x + 18) + 4 x$

Group like terms:

$7 x + 4 = (- 4 x + 4 x) + 18$

Simplify the arithmetic:

$7 x + 4 = 18$

Subtract from both sides:

$(7 x + 4) - 4 = 18 - 4$

Simplify the arithmetic:

$7 x = 18 - 4$

Simplify the arithmetic:

$7 x = 14$

Divide both sides by :

$\frac{(7 x)}{7} = \frac{14}{7}$

Simplify the fraction:

$x = \frac{14}{7}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 7)}{(1 \cdot 7)}$

Factor out and cancel the greatest common factor:

$x = 2$

3. List the solutions

$x = 22, 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 3 x + 4 |$
$y = 2 | 2 x - 9 |$
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 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

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