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Solution - Absolute value equations

Exact form:

x = \frac{7}{16}, - \frac{1}{2}

x=\frac{7}{16} , -\frac{1}{2}

Decimal form:

x = 0.438, - 0.5

x=0.438 , -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
$3 | 3 x - 1 | = | - 7 x + 4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| 3 x - 1 \| = \| - 7 x + 4 \|$
$x = + y$	$3 (3 x - 1) = (- 7 x + 4)$
$x = - y$	$3 (3 x - 1) = - (- 7 x + 4)$
$+ x = y$	$3 (3 x - 1) = (- 7 x + 4)$
$- x = y$	$3 (- (3 x - 1)) = (- 7 x + 4)$

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:

$\| x \| = \| y \|$	$3 \| 3 x - 1 \| = \| - 7 x + 4 \|$
$x = + y$ , $+ x = y$	$3 (3 x - 1) = (- 7 x + 4)$
$x = - y$ , $- x = y$	$3 (3 x - 1) = - (- 7 x + 4)$

2. Solve the two equations for x

12 additional steps

$3 \cdot (3 x - 1) = (- 7 x + 4)$

Expand the parentheses:

$3 \cdot 3 x + 3 \cdot - 1 = (- 7 x + 4)$

Multiply the coefficients:

$9 x + 3 \cdot - 1 = (- 7 x + 4)$

Simplify the arithmetic:

$9 x - 3 = (- 7 x + 4)$

Add to both sides:

$(9 x - 3) + 7 x = (- 7 x + 4) + 7 x$

Group like terms:

$(9 x + 7 x) - 3 = (- 7 x + 4) + 7 x$

Simplify the arithmetic:

$16 x - 3 = (- 7 x + 4) + 7 x$

Group like terms:

$16 x - 3 = (- 7 x + 7 x) + 4$

Simplify the arithmetic:

$16 x - 3 = 4$

Add to both sides:

$(16 x - 3) + 3 = 4 + 3$

Simplify the arithmetic:

$16 x = 4 + 3$

Simplify the arithmetic:

$16 x = 7$

Divide both sides by :

$\frac{(16 x)}{16} = \frac{7}{16}$

Simplify the fraction:

$x = \frac{7}{16}$

13 additional steps

$3 \cdot (3 x - 1) = - (- 7 x + 4)$

Expand the parentheses:

$3 \cdot 3 x + 3 \cdot - 1 = - (- 7 x + 4)$

Multiply the coefficients:

$9 x + 3 \cdot - 1 = - (- 7 x + 4)$

Simplify the arithmetic:

$9 x - 3 = - (- 7 x + 4)$

Expand the parentheses:

$9 x - 3 = 7 x - 4$

Subtract from both sides:

$(9 x - 3) - 7 x = (7 x - 4) - 7 x$

Group like terms:

$(9 x - 7 x) - 3 = (7 x - 4) - 7 x$

Simplify the arithmetic:

$2 x - 3 = (7 x - 4) - 7 x$

Group like terms:

$2 x - 3 = (7 x - 7 x) - 4$

Simplify the arithmetic:

$2 x - 3 = - 4$

Add to both sides:

$(2 x - 3) + 3 = - 4 + 3$

Simplify the arithmetic:

$2 x = - 4 + 3$

Simplify the arithmetic:

$2 x = - 1$

Divide both sides by :

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

Simplify the fraction:

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

3. List the solutions

$x = \frac{7}{16}, - \frac{1}{2}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = 3 | 3 x - 1 |$
$y = | - 7 x + 4 |$
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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