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Solution - Absolute value equations

Exact form:

x = - 3, 1

x=-3 , 1

See steps

Step-by-step explanation

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

$| 0.5 x - 0.7 | - | 0.6 x - 0.4 | = 0$

Add $| 0.6 x - 0.4 |$ to both sides of the equation:

$| 0.5 x - 0.7 | - | 0.6 x - 0.4 | + | 0.6 x - 0.4 | = | 0.6 x - 0.4 |$

Simplify the arithmetic

$| 0.5 x - 0.7 | = | 0.6 x - 0.4 |$

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
$| 0.5 x - 0.7 | = | 0.6 x - 0.4 |$
without the absolute value bars:

$\| x \| = \| y \|$	$\| 0.5 x - 0.7 \| = \| 0.6 x - 0.4 \|$
$x = + y$	$(0.5 x - 0.7) = (0.6 x - 0.4)$
$x = - y$	$(0.5 x - 0.7) = (- (0.6 x - 0.4))$
$+ x = y$	$(0.5 x - 0.7) = (0.6 x - 0.4)$
$- x = y$	$- (0.5 x - 0.7) = (0.6 x - 0.4)$

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:

$\| x \| = \| y \|$	$\| 0.5 x - 0.7 \| = \| 0.6 x - 0.4 \|$
$x = + y$ , $+ x = y$	$(0.5 x - 0.7) = (0.6 x - 0.4)$
$x = - y$ , $- x = y$	$(0.5 x - 0.7) = (- (0.6 x - 0.4))$

3. Solve the two equations for x

12 additional steps

$(0.5 x - 0.7) = (0.6 x - 0.4)$

Subtract from both sides:

$(0.5 x - 0.7) - 0.6 x = (0.6 x - 0.4) - 0.6 x$

Group like terms:

$(0.5 x - 0.6 x) - 0.7 = (0.6 x - 0.4) - 0.6 x$

Simplify the arithmetic:

$- 0.1 x - 0.7 = (0.6 x - 0.4) - 0.6 x$

Group like terms:

$- 0.1 x - 0.7 = (0.6 x - 0.6 x) - 0.4$

Simplify the arithmetic:

$- 0.1 x - 0.7 = - 0.4$

Add to both sides:

$(- 0.1 x - 0.7) + 0.7 = - 0.4 + 0.7$

Simplify the arithmetic:

$- 0.1 x = - 0.4 + 0.7$

Simplify the arithmetic:

$- 0.1 x = 0.3$

Divide both sides by :

$\frac{(- 0.1 x)}{- 0.1} = \frac{0.3}{- 0.1}$

Cancel out the negatives:

$\frac{0.1 x}{0.1} = \frac{0.3}{- 0.1}$

Simplify the arithmetic:

$x = \frac{0.3}{- 0.1}$

Move the negative sign from the denominator to the numerator:

$x = \frac{- 0.3}{0.1}$

Simplify the arithmetic:

$x = - 3$

11 additional steps

$(0.5 x - 0.7) = - (0.6 x - 0.4)$

Expand the parentheses:

$(0.5 x - 0.7) = - 0.6 x + 0.4$

Add to both sides:

$(0.5 x - 0.7) + 0.6 x = (- 0.6 x + 0.4) + 0.6 x$

Group like terms:

$(0.5 x + 0.6 x) - 0.7 = (- 0.6 x + 0.4) + 0.6 x$

Simplify the arithmetic:

$1.1 x - 0.7 = (- 0.6 x + 0.4) + 0.6 x$

Group like terms:

$1.1 x - 0.7 = (- 0.6 x + 0.6 x) + 0.4$

Simplify the arithmetic:

$1.1 x - 0.7 = 0.4$

Add to both sides:

$(1.1 x - 0.7) + 0.7 = 0.4 + 0.7$

Simplify the arithmetic:

$1.1 x = 0.4 + 0.7$

Simplify the arithmetic:

$1.1 x = 1.1$

Divide both sides by :

$\frac{(1.1 x)}{1.1} = \frac{1.1}{1.1}$

Simplify the arithmetic:

$x = \frac{1.1}{1.1}$

Cancel terms:

$x = 1$

4. List the solutions

$x = - 3, 1$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = | 0.5 x - 0.7 |$
$y = | 0.6 x - 0.4 |$
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 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