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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
$| 4 x - 10 | = 2 | 2 x + 3 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 4 x - 10 \| = 2 \| 2 x + 3 \|$
$x = + y$	$(4 x - 10) = 2 (2 x + 3)$
$x = - y$	$(4 x - 10) = 2 (- (2 x + 3))$
$+ x = y$	$(4 x - 10) = 2 (2 x + 3)$
$- x = y$	$- (4 x - 10) = 2 (2 x + 3)$

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:

$\| x \| = \| y \|$	$\| 4 x - 10 \| = 2 \| 2 x + 3 \|$
$x = + y$ , $+ x = y$	$(4 x - 10) = 2 (2 x + 3)$
$x = - y$ , $- x = y$	$(4 x - 10) = 2 (- (2 x + 3))$

2. Solve the two equations for x

8 additional steps

$(4 x - 10) = 2 \cdot (2 x + 3)$

Expand the parentheses:

$(4 x - 10) = 2 \cdot 2 x + 2 \cdot 3$

Multiply the coefficients:

$(4 x - 10) = 4 x + 2 \cdot 3$

Simplify the arithmetic:

$(4 x - 10) = 4 x + 6$

Subtract from both sides:

$(4 x - 10) - 4 x = (4 x + 6) - 4 x$

Group like terms:

$(4 x - 4 x) - 10 = (4 x + 6) - 4 x$

Simplify the arithmetic:

$- 10 = (4 x + 6) - 4 x$

Group like terms:

$- 10 = (4 x - 4 x) + 6$

Simplify the arithmetic:

$- 10 = 6$

The statement is false:

$- 10 = 6$

The equation is false so it has no solution.

15 additional steps

$(4 x - 10) = 2 \cdot (- (2 x + 3))$

Expand the parentheses:

$(4 x - 10) = 2 \cdot (- 2 x - 3)$

Expand the parentheses:

$(4 x - 10) = 2 \cdot - 2 x + 2 \cdot - 3$

Multiply the coefficients:

$(4 x - 10) = - 4 x + 2 \cdot - 3$

Simplify the arithmetic:

$(4 x - 10) = - 4 x - 6$

Add to both sides:

$(4 x - 10) + 4 x = (- 4 x - 6) + 4 x$

Group like terms:

$(4 x + 4 x) - 10 = (- 4 x - 6) + 4 x$

Simplify the arithmetic:

$8 x - 10 = (- 4 x - 6) + 4 x$

Group like terms:

$8 x - 10 = (- 4 x + 4 x) - 6$

Simplify the arithmetic:

$8 x - 10 = - 6$

Add to both sides:

$(8 x - 10) + 10 = - 6 + 10$

Simplify the arithmetic:

$8 x = - 6 + 10$

Simplify the arithmetic:

$8 x = 4$

Divide both sides by :

$\frac{(8 x)}{8} = \frac{4}{8}$

Simplify the fraction:

$x = \frac{4}{8}$

Find the greatest common factor of the numerator and denominator:

$x = \frac{(1 \cdot 4)}{(2 \cdot 4)}$

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:
$y = | 4 x - 10 |$
$y = 2 | 2 x + 3 |$
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