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Solution - Absolute value equations

Exact form:

x = 3, \frac{2}{3}

x=3 , \frac{2}{3}

Decimal form:

x = 3, 0.667

x=3 , 0.667

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
$| 4 x - 5 | = | 2 x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 5 \| = \| 2 x + 1 \|$
$x = + y$	$(4 x - 5) = (2 x + 1)$
$x = - y$	$(4 x - 5) = - (2 x + 1)$
$+ x = y$	$(4 x - 5) = (2 x + 1)$
$- x = y$	$- (4 x - 5) = (2 x + 1)$

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:

$\| x \| = \| y \|$	$\| 4 x - 5 \| = \| 2 x + 1 \|$
$x = + y$ , $+ x = y$	$(4 x - 5) = (2 x + 1)$
$x = - y$ , $- x = y$	$(4 x - 5) = - (2 x + 1)$

2. Solve the two equations for x

11 additional steps

$(4 x - 5) = (2 x + 1)$

Subtract from both sides:

$(4 x - 5) - 2 x = (2 x + 1) - 2 x$

Group like terms:

$(4 x - 2 x) - 5 = (2 x + 1) - 2 x$

Simplify the arithmetic:

$2 x - 5 = (2 x + 1) - 2 x$

Group like terms:

$2 x - 5 = (2 x - 2 x) + 1$

Simplify the arithmetic:

$2 x - 5 = 1$

Add to both sides:

$(2 x - 5) + 5 = 1 + 5$

Simplify the arithmetic:

$2 x = 1 + 5$

Simplify the arithmetic:

$2 x = 6$

Divide both sides by :

$\frac{(2 x)}{2} = \frac{6}{2}$

Simplify the fraction:

$x = \frac{6}{2}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(3 \cdot 2)}{(1 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = 3$

12 additional steps

$(4 x - 5) = - (2 x + 1)$

Expand the parentheses:

$(4 x - 5) = - 2 x - 1$

Add to both sides:

$(4 x - 5) + 2 x = (- 2 x - 1) + 2 x$

Group like terms:

$(4 x + 2 x) - 5 = (- 2 x - 1) + 2 x$

Simplify the arithmetic:

$6 x - 5 = (- 2 x - 1) + 2 x$

Group like terms:

$6 x - 5 = (- 2 x + 2 x) - 1$

Simplify the arithmetic:

$6 x - 5 = - 1$

Add to both sides:

$(6 x - 5) + 5 = - 1 + 5$

Simplify the arithmetic:

$6 x = - 1 + 5$

Simplify the arithmetic:

$6 x = 4$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{4}{6}$

Simplify the fraction:

$x = \frac{4}{6}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(2 \cdot 2)}{(3 \cdot 2)}$

Factor out and cancel the greatest common factor:

$x = \frac{2}{3}$

3. List the solutions

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

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 5 |$
$y = | 2 x + 1 |$
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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