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Solution - Absolute value equations

Exact form:

x = \frac{7}{5}, 17

x=\frac{7}{5} , 17

Mixed number form:

x = 1 \frac{2}{5}, 17

x=1\frac{2}{5} , 17

Decimal form:

x = 1.4, 17

x=1.4 , 17

See steps

Step-by-step explanation

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

$3 | x - 4 | + | 2 x + 5 | = 0$

Add $- | 2 x + 5 |$ to both sides of the equation:

$3 | x - 4 | + | 2 x + 5 | - | 2 x + 5 | = - | 2 x + 5 |$

Simplify the arithmetic

$3 | x - 4 | = - | 2 x + 5 |$

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
$3 | x - 4 | = - | 2 x + 5 |$
without the absolute value bars:

$\| x \| = \| y \|$	$3 \| x - 4 \| = - \| 2 x + 5 \|$
$x = + y$	$3 (x - 4) = - (2 x + 5)$
$x = - y$	$3 (x - 4) = - - (2 x + 5)$
$+ x = y$	$3 (x - 4) = - (2 x + 5)$
$- x = y$	$3 (- (x - 4)) = - (2 x + 5)$

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:

$\| x \| = \| y \|$	$3 \| x - 4 \| = - \| 2 x + 5 \|$
$x = + y$ , $+ x = y$	$3 (x - 4) = - (2 x + 5)$
$x = - y$ , $- x = y$	$3 (x - 4) = - - (2 x + 5)$

3. Solve the two equations for x

12 additional steps

$3 \cdot (x - 4) = - (2 x + 5)$

Expand the parentheses:

$3 x + 3 \cdot - 4 = - (2 x + 5)$

Simplify the arithmetic:

$3 x - 12 = - (2 x + 5)$

Expand the parentheses:

$3 x - 12 = - 2 x - 5$

Add to both sides:

$(3 x - 12) + 2 x = (- 2 x - 5) + 2 x$

Group like terms:

$(3 x + 2 x) - 12 = (- 2 x - 5) + 2 x$

Simplify the arithmetic:

$5 x - 12 = (- 2 x - 5) + 2 x$

Group like terms:

$5 x - 12 = (- 2 x + 2 x) - 5$

Simplify the arithmetic:

$5 x - 12 = - 5$

Add to both sides:

$(5 x - 12) + 12 = - 5 + 12$

Simplify the arithmetic:

$5 x = - 5 + 12$

Simplify the arithmetic:

$5 x = 7$

Divide both sides by :

$\frac{(5 x)}{5} = \frac{7}{5}$

Simplify the fraction:

$x = \frac{7}{5}$

10 additional steps

$3 \cdot (x - 4) = - (- (2 x + 5))$

Expand the parentheses:

$3 x + 3 \cdot - 4 = - (- (2 x + 5))$

Simplify the arithmetic:

$3 x - 12 = - (- (2 x + 5))$

Resolve the double minus:

$3 x - 12 = 2 x + 5$

Subtract from both sides:

$(3 x - 12) - 2 x = (2 x + 5) - 2 x$

Group like terms:

$(3 x - 2 x) - 12 = (2 x + 5) - 2 x$

Simplify the arithmetic:

$x - 12 = (2 x + 5) - 2 x$

Group like terms:

$x - 12 = (2 x - 2 x) + 5$

Simplify the arithmetic:

$x - 12 = 5$

Add to both sides:

$(x - 12) + 12 = 5 + 12$

Simplify the arithmetic:

$x = 5 + 12$

Simplify the arithmetic:

$x = 17$

4. List the solutions

$x = \frac{7}{5}, 17$
(2 solution(s))

5. Graph

Each line represents the function of one side of the equation:
$y = 3 | x - 4 |$
$y = - | 2 x + 5 |$
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