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Solution - Absolute value equations

Exact form:

x = \frac{29}{4}, \frac{11}{14}

x=\frac{29}{4} , \frac{11}{14}

Mixed number form:

x = 7 \frac{1}{4}, \frac{11}{14}

x=7\frac{1}{4} , \frac{11}{14}

Decimal form:

x = 7.25, 0.786

x=7.25 , 0.786

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
$| 9 x - 20 | = | 5 x + 9 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x - 20 \| = \| 5 x + 9 \|$
$x = + y$	$(9 x - 20) = (5 x + 9)$
$x = - y$	$(9 x - 20) = - (5 x + 9)$
$+ x = y$	$(9 x - 20) = (5 x + 9)$
$- x = y$	$- (9 x - 20) = (5 x + 9)$

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:

$\| x \| = \| y \|$	$\| 9 x - 20 \| = \| 5 x + 9 \|$
$x = + y$ , $+ x = y$	$(9 x - 20) = (5 x + 9)$
$x = - y$ , $- x = y$	$(9 x - 20) = - (5 x + 9)$

2. Solve the two equations for x

9 additional steps

$(9 x - 20) = (5 x + 9)$

Subtract from both sides:

$(9 x - 20) - 5 x = (5 x + 9) - 5 x$

Group like terms:

$(9 x - 5 x) - 20 = (5 x + 9) - 5 x$

Simplify the arithmetic:

$4 x - 20 = (5 x + 9) - 5 x$

Group like terms:

$4 x - 20 = (5 x - 5 x) + 9$

Simplify the arithmetic:

$4 x - 20 = 9$

Add to both sides:

$(4 x - 20) + 20 = 9 + 20$

Simplify the arithmetic:

$4 x = 9 + 20$

Simplify the arithmetic:

$4 x = 29$

Divide both sides by :

$\frac{(4 x)}{4} = \frac{29}{4}$

Simplify the fraction:

$x = \frac{29}{4}$

10 additional steps

$(9 x - 20) = - (5 x + 9)$

Expand the parentheses:

$(9 x - 20) = - 5 x - 9$

Add to both sides:

$(9 x - 20) + 5 x = (- 5 x - 9) + 5 x$

Group like terms:

$(9 x + 5 x) - 20 = (- 5 x - 9) + 5 x$

Simplify the arithmetic:

$14 x - 20 = (- 5 x - 9) + 5 x$

Group like terms:

$14 x - 20 = (- 5 x + 5 x) - 9$

Simplify the arithmetic:

$14 x - 20 = - 9$

Add to both sides:

$(14 x - 20) + 20 = - 9 + 20$

Simplify the arithmetic:

$14 x = - 9 + 20$

Simplify the arithmetic:

$14 x = 11$

Divide both sides by :

$\frac{(14 x)}{14} = \frac{11}{14}$

Simplify the fraction:

$x = \frac{11}{14}$

3. List the solutions

$x = \frac{29}{4}, \frac{11}{14}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 x - 20 |$
$y = | 5 x + 9 |$
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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