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Solution - Absolute value equations

Exact form:

x = \frac{1}{19}, 5

x=\frac{1}{19} , 5

Decimal form:

x = 0.053, 5

x=0.053 , 5

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
$| 9 x + 2 | = | - 10 x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 9 x + 2 \| = \| - 10 x + 3 \|$
$x = + y$	$(9 x + 2) = (- 10 x + 3)$
$x = - y$	$(9 x + 2) = - (- 10 x + 3)$
$+ x = y$	$(9 x + 2) = (- 10 x + 3)$
$- x = y$	$- (9 x + 2) = (- 10 x + 3)$

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:

$\| x \| = \| y \|$	$\| 9 x + 2 \| = \| - 10 x + 3 \|$
$x = + y$ , $+ x = y$	$(9 x + 2) = (- 10 x + 3)$
$x = - y$ , $- x = y$	$(9 x + 2) = - (- 10 x + 3)$

2. Solve the two equations for x

9 additional steps

$(9 x + 2) = (- 10 x + 3)$

Add to both sides:

$(9 x + 2) + 10 x = (- 10 x + 3) + 10 x$

Group like terms:

$(9 x + 10 x) + 2 = (- 10 x + 3) + 10 x$

Simplify the arithmetic:

$19 x + 2 = (- 10 x + 3) + 10 x$

Group like terms:

$19 x + 2 = (- 10 x + 10 x) + 3$

Simplify the arithmetic:

$19 x + 2 = 3$

Subtract from both sides:

$(19 x + 2) - 2 = 3 - 2$

Simplify the arithmetic:

$19 x = 3 - 2$

Simplify the arithmetic:

$19 x = 1$

Divide both sides by :

$\frac{(19 x)}{19} = \frac{1}{19}$

Simplify the fraction:

$x = \frac{1}{19}$

11 additional steps

$(9 x + 2) = - (- 10 x + 3)$

Expand the parentheses:

$(9 x + 2) = 10 x - 3$

Subtract from both sides:

$(9 x + 2) - 10 x = (10 x - 3) - 10 x$

Group like terms:

$(9 x - 10 x) + 2 = (10 x - 3) - 10 x$

Simplify the arithmetic:

$- x + 2 = (10 x - 3) - 10 x$

Group like terms:

$- x + 2 = (10 x - 10 x) - 3$

Simplify the arithmetic:

$- x + 2 = - 3$

Subtract from both sides:

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

Simplify the arithmetic:

$- x = - 3 - 2$

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:

$x = 5$

3. List the solutions

$x = \frac{1}{19}, 5$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 9 x + 2 |$
$y = | - 10 x + 3 |$
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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