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Solution - Absolute value equations

Exact form:

x = 16, - 2

x=16 , -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
$| 4 x - 1 | = 3 | x + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 1 \| = 3 \| x + 5 \|$
$x = + y$	$(4 x - 1) = 3 (x + 5)$
$x = - y$	$(4 x - 1) = 3 (- (x + 5))$
$+ x = y$	$(4 x - 1) = 3 (x + 5)$
$- x = y$	$- (4 x - 1) = 3 (x + 5)$

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:

$\| x \| = \| y \|$	$\| 4 x - 1 \| = 3 \| x + 5 \|$
$x = + y$ , $+ x = y$	$(4 x - 1) = 3 (x + 5)$
$x = - y$ , $- x = y$	$(4 x - 1) = 3 (- (x + 5))$

2. Solve the two equations for x

9 additional steps

$(4 x - 1) = 3 \cdot (x + 5)$

Expand the parentheses:

$(4 x - 1) = 3 x + 3 \cdot 5$

Simplify the arithmetic:

$(4 x - 1) = 3 x + 15$

Subtract from both sides:

$(4 x - 1) - 3 x = (3 x + 15) - 3 x$

Group like terms:

$(4 x - 3 x) - 1 = (3 x + 15) - 3 x$

Simplify the arithmetic:

$x - 1 = (3 x + 15) - 3 x$

Group like terms:

$x - 1 = (3 x - 3 x) + 15$

Simplify the arithmetic:

$x - 1 = 15$

Add to both sides:

$(x - 1) + 1 = 15 + 1$

Simplify the arithmetic:

$x = 15 + 1$

Simplify the arithmetic:

$x = 16$

16 additional steps

$(4 x - 1) = 3 \cdot (- (x + 5))$

Expand the parentheses:

$(4 x - 1) = 3 \cdot (- x - 5)$

$(4 x - 1) = 3 \cdot - x + 3 \cdot - 5$

Group like terms:

$(4 x - 1) = (3 \cdot - 1) x + 3 \cdot - 5$

Multiply the coefficients:

$(4 x - 1) = - 3 x + 3 \cdot - 5$

Simplify the arithmetic:

$(4 x - 1) = - 3 x - 15$

Add to both sides:

$(4 x - 1) + 3 x = (- 3 x - 15) + 3 x$

Group like terms:

$(4 x + 3 x) - 1 = (- 3 x - 15) + 3 x$

Simplify the arithmetic:

$7 x - 1 = (- 3 x - 15) + 3 x$

Group like terms:

$7 x - 1 = (- 3 x + 3 x) - 15$

Simplify the arithmetic:

$7 x - 1 = - 15$

Add to both sides:

$(7 x - 1) + 1 = - 15 + 1$

Simplify the arithmetic:

$7 x = - 15 + 1$

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

$x = 16, - 2$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 1 |$
$y = 3 | x + 5 |$
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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