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Solution - Absolute value equations

Exact form:

x = - \frac{1}{16}

x=-\frac{1}{16}

Decimal form:

x = - 0.062

x=-0.062

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
$| 8 x - 3 | = 4 | 2 x + 1 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 8 x - 3 \| = 4 \| 2 x + 1 \|$
$x = + y$	$(8 x - 3) = 4 (2 x + 1)$
$x = - y$	$(8 x - 3) = 4 (- (2 x + 1))$
$+ x = y$	$(8 x - 3) = 4 (2 x + 1)$
$- x = y$	$- (8 x - 3) = 4 (2 x + 1)$

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:

$\| x \| = \| y \|$	$\| 8 x - 3 \| = 4 \| 2 x + 1 \|$
$x = + y$ , $+ x = y$	$(8 x - 3) = 4 (2 x + 1)$
$x = - y$ , $- x = y$	$(8 x - 3) = 4 (- (2 x + 1))$

2. Solve the two equations for x

8 additional steps

$(8 x - 3) = 4 \cdot (2 x + 1)$

Expand the parentheses:

$(8 x - 3) = 4 \cdot 2 x + 4 \cdot 1$

Multiply the coefficients:

$(8 x - 3) = 8 x + 4 \cdot 1$

Simplify the arithmetic:

$(8 x - 3) = 8 x + 4$

Subtract from both sides:

$(8 x - 3) - 8 x = (8 x + 4) - 8 x$

Group like terms:

$(8 x - 8 x) - 3 = (8 x + 4) - 8 x$

Simplify the arithmetic:

$- 3 = (8 x + 4) - 8 x$

Group like terms:

$- 3 = (8 x - 8 x) + 4$

Simplify the arithmetic:

$- 3 = 4$

The statement is false:

$- 3 = 4$

The equation is false so it has no solution.

13 additional steps

$(8 x - 3) = 4 \cdot (- (2 x + 1))$

Expand the parentheses:

$(8 x - 3) = 4 \cdot (- 2 x - 1)$

Expand the parentheses:

$(8 x - 3) = 4 \cdot - 2 x + 4 \cdot - 1$

Multiply the coefficients:

$(8 x - 3) = - 8 x + 4 \cdot - 1$

Simplify the arithmetic:

$(8 x - 3) = - 8 x - 4$

Add to both sides:

$(8 x - 3) + 8 x = (- 8 x - 4) + 8 x$

Group like terms:

$(8 x + 8 x) - 3 = (- 8 x - 4) + 8 x$

Simplify the arithmetic:

$16 x - 3 = (- 8 x - 4) + 8 x$

Group like terms:

$16 x - 3 = (- 8 x + 8 x) - 4$

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:

$16 x = - 1$

Divide both sides by :

$\frac{(16 x)}{16} = \frac{- 1}{16}$

Simplify the fraction:

$x = \frac{- 1}{16}$

3. Graph

Each line represents the function of one side of the equation:
$y = | 8 x - 3 |$
$y = 4 | 2 x + 1 |$
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 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