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Solution - Absolute value equations

Exact form:

x = \frac{7}{19}, \frac{5}{23}

x=\frac{7}{19} , \frac{5}{23}

Decimal form:

x = 0.368, 0.217

x=0.368 , 0.217

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
$| 2 x + 1 | = 3 | 7 x - 2 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 2 x + 1 \| = 3 \| 7 x - 2 \|$
$x = + y$	$(2 x + 1) = 3 (7 x - 2)$
$x = - y$	$(2 x + 1) = 3 (- (7 x - 2))$
$+ x = y$	$(2 x + 1) = 3 (7 x - 2)$
$- x = y$	$- (2 x + 1) = 3 (7 x - 2)$

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:

$\| x \| = \| y \|$	$\| 2 x + 1 \| = 3 \| 7 x - 2 \|$
$x = + y$ , $+ x = y$	$(2 x + 1) = 3 (7 x - 2)$
$x = - y$ , $- x = y$	$(2 x + 1) = 3 (- (7 x - 2))$

2. Solve the two equations for x

14 additional steps

$(2 x + 1) = 3 \cdot (7 x - 2)$

Expand the parentheses:

$(2 x + 1) = 3 \cdot 7 x + 3 \cdot - 2$

Multiply the coefficients:

$(2 x + 1) = 21 x + 3 \cdot - 2$

Simplify the arithmetic:

$(2 x + 1) = 21 x - 6$

Subtract from both sides:

$(2 x + 1) - 21 x = (21 x - 6) - 21 x$

Group like terms:

$(2 x - 21 x) + 1 = (21 x - 6) - 21 x$

Simplify the arithmetic:

$- 19 x + 1 = (21 x - 6) - 21 x$

Group like terms:

$- 19 x + 1 = (21 x - 21 x) - 6$

Simplify the arithmetic:

$- 19 x + 1 = - 6$

Subtract from both sides:

$(- 19 x + 1) - 1 = - 6 - 1$

Simplify the arithmetic:

$- 19 x = - 6 - 1$

Simplify the arithmetic:

$- 19 x = - 7$

Divide both sides by :

$\frac{(- 19 x)}{- 19} = \frac{- 7}{- 19}$

Cancel out the negatives:

$\frac{19 x}{19} = \frac{- 7}{- 19}$

Simplify the fraction:

$x = \frac{- 7}{- 19}$

Cancel out the negatives:

$x = \frac{7}{19}$

13 additional steps

$(2 x + 1) = 3 \cdot (- (7 x - 2))$

Expand the parentheses:

$(2 x + 1) = 3 \cdot (- 7 x + 2)$

Expand the parentheses:

$(2 x + 1) = 3 \cdot - 7 x + 3 \cdot 2$

Multiply the coefficients:

$(2 x + 1) = - 21 x + 3 \cdot 2$

Simplify the arithmetic:

$(2 x + 1) = - 21 x + 6$

Add to both sides:

$(2 x + 1) + 21 x = (- 21 x + 6) + 21 x$

Group like terms:

$(2 x + 21 x) + 1 = (- 21 x + 6) + 21 x$

Simplify the arithmetic:

$23 x + 1 = (- 21 x + 6) + 21 x$

Group like terms:

$23 x + 1 = (- 21 x + 21 x) + 6$

Simplify the arithmetic:

$23 x + 1 = 6$

Subtract from both sides:

$(23 x + 1) - 1 = 6 - 1$

Simplify the arithmetic:

$23 x = 6 - 1$

Simplify the arithmetic:

$23 x = 5$

Divide both sides by :

$\frac{(23 x)}{23} = \frac{5}{23}$

Simplify the fraction:

$x = \frac{5}{23}$

3. List the solutions

$x = \frac{7}{19}, \frac{5}{23}$
(2 solution(s))

4. Graph

Each line represents the function of one side of the equation:
$y = | 2 x + 1 |$
$y = 3 | 7 x - 2 |$
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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