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Solution - Absolute value equations

Exact form:

x = - \frac{1}{2}

x=-\frac{1}{2}

Decimal form:

x = - 0.5

x=-0.5

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
$3 | x | = 3 | x + 1 |$
without the absolute value bars:

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

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:

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

2. Solve the two equations for x

6 additional steps

$3 x = 3 \cdot (x + 1)$

Expand the parentheses:

$3 x = 3 x + 3 \cdot 1$

Simplify the arithmetic:

$3 x = 3 x + 3$

Subtract from both sides:

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

Simplify the arithmetic:

$0 = (3 x + 3) - 3 x$

Group like terms:

$0 = (3 x - 3 x) + 3$

Simplify the arithmetic:

$0 = 3$

The statement is false:

$0 = 3$

The equation is false so it has no solution.

12 additional steps

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

Expand the parentheses:

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

$3 x = 3 \cdot - x + 3 \cdot - 1$

Group like terms:

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

Multiply the coefficients:

$3 x = - 3 x + 3 \cdot - 1$

Simplify the arithmetic:

$3 x = - 3 x - 3$

Add to both sides:

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

Simplify the arithmetic:

$6 x = (- 3 x - 3) + 3 x$

Group like terms:

$6 x = (- 3 x + 3 x) - 3$

Simplify the arithmetic:

$6 x = - 3$

Divide both sides by :

$\frac{(6 x)}{6} = \frac{- 3}{6}$

Simplify the fraction:

$x = \frac{- 3}{6}$

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = \frac{- 1}{2}$

3. Graph

Each line represents the function of one side of the equation:
$y = 3 | x |$
$y = 3 | x + 1 |$
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. 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. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved