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Solution - Absolute value equations

Exact form:

x = 3, \frac{1}{3}

x=3 , \frac{1}{3}

Decimal form:

x = 3, 0.333

x=3 , 0.333

See steps

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

$| 4 x - 4 | - 2 | x + 1 | = 0$

Add $2 | x + 1 |$ to both sides of the equation:

$| 4 x - 4 | - 2 | x + 1 | + 2 | x + 1 | = 2 | x + 1 |$

Simplify the arithmetic

$| 4 x - 4 | = 2 | x + 1 |$

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

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

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:

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

3. Solve the two equations for x

13 additional steps

$(4 x - 4) = 2 \cdot (x + 1)$

Expand the parentheses:

$(4 x - 4) = 2 x + 2 \cdot 1$

Simplify the arithmetic:

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

Subtract from both sides:

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

Group like terms:

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

Simplify the arithmetic:

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

Group like terms:

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

Simplify the arithmetic:

$2 x - 4 = 2$

Add to both sides:

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

Simplify the arithmetic:

$2 x = 2 + 4$

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:

$x = 3$

16 additional steps

$(4 x - 4) = 2 \cdot (- (x + 1))$

Expand the parentheses:

$(4 x - 4) = 2 \cdot (- x - 1)$

$(4 x - 4) = 2 \cdot - x + 2 \cdot - 1$

Group like terms:

$(4 x - 4) = (2 \cdot - 1) x + 2 \cdot - 1$

Multiply the coefficients:

$(4 x - 4) = - 2 x + 2 \cdot - 1$

Simplify the arithmetic:

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

Add to both sides:

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

Group like terms:

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

Simplify the arithmetic:

$6 x - 4 = (- 2 x - 2) + 2 x$

Group like terms:

$6 x - 4 = (- 2 x + 2 x) - 2$

Simplify the arithmetic:

$6 x - 4 = - 2$

Add to both sides:

$(6 x - 4) + 4 = - 2 + 4$

Simplify the arithmetic:

$6 x = - 2 + 4$

Simplify the arithmetic:

$6 x = 2$

Divide both sides by :

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

Simplify the fraction:

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

Find the greatest common factor of the numerator and denominator:

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

Factor out and cancel the greatest common factor:

$x = \frac{1}{3}$

4. List the solutions

$x = 3, \frac{1}{3}$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 4 x - 4 |$
$y = 2 | x + 1 |$
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 with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Solution - Absolute value equations

Step-by-step explanation

1. Rewrite the equation with one absolute value terms on each side

2. Rewrite the equation without absolute value bars

3. Solve the two equations for x

4. List the solutions

5. Graph

Why learn this

Terms and topics

Related links

Latest Related Drills Solved