Absolute Value Definition




Steps for Solving Linear Absolute Value Equations: i.e. \(|ax + b| = c\)

1. Isolate the absolute value.

2. Identify what the isolated absolute value is set equal to…

• If the absolute value is set equal to zero, remove absolute value symbols & solve the equation to get one solution.
• If the absolute value is set equal to a negative number, there is no solution.
• If the absolute value is set equal to a positive number, set the argument (expression within the absolute value) equal to the number and set it equal to the opposite of the number, using an ‘or’ statement in between the two equations. Then solve each equation separately to get two solutions.

Examples

\(|3x+12|+7=7\)
\(|3x+12|=0\)
Because this equal to zero, we have ONE solution

\(|3x-7|+7=2\)
\(|3x-7|=-5\)
Because this is equal to a negative number, we have NO solution

\(|3x-7|+7=9\)
\(|3x-7|=2\)
Because this is equal to a positive number, we have TWO solution


Steps for Solving Linear Absolute Value Inequalities: i.e. \(|ax+b| \leq c\)

1. Isolate the absolute value.
2. Identify what the absolute value inequality is set “equal” to…

ZERO

a. If the absolute value is less than zero, there is no solution.
b. If the absolute value is less than or equal to zero, there is one solution. Just set the argument equal to zero and solve.
c. If the absolute value is greater than or equal to zero, the solution is all real numbers.
d. If the absolute value is greater than zero, the solution is all real numbers except for the value which makes it equal to zero. This will be written as a union.

NEGATIVE Number

e. If the absolute value is less than or less than or equal to a negative number, there is no solution. The absolute value of something will never be less than or equal to a negative number.
f. If the absolute value is greater than or greater than or equal to a negative number, the solution is all real numbers. The absolute value of something will always be greater than a negative number.


POSITIVE Number

g. If the absolute value is less than or less than or equal to a positive number, the problem can be approached two ways. Either way, the solution will be written as an intersection.

i. Place the argument in a 3-part inequality (compound) between the opposite of the number and the number, then solve.
ii. Set the argument less than the number and greater than the opposite of the number using an “and” statement in between the two inequalities.

If the absolute value is greater than or greater than or equal to a positive number, set the argument less than the opposite of the number and greater than the number using an ‘or’ statement in between the two inequalities. Then solve each inequality, writing the solution as a union of the two solutions.


RELATIONSHIP BETWEEN SQUARE ROOTS AND ABSOLUTE VALUES

Recall from algebra that a number is called a square root of \(\sqrt{x}\) if its square is \(\sqrt{x}\). Recall also that every positive real number has two square roots, one positive and one negative; the positive square root is denoted by \(\sqrt{x}\) and the negative square root by \(-\sqrt{x}\). For example, the positive square root of 9 is \(\sqrt{9} = 3\), and the negative square root of 9 is \(− \sqrt{9} = −3\).

NOTE: Readers who may have been taught to write √9 as ±3 should stop doing so, since it is incorrect.


It is a common error to replace \(\sqrt{x^2}\) by x. Although this is correct when x is nonnegative, it is false for negative x. For example, if \(x = −4\), then

\(\sqrt{x^2}=\sqrt{(-4)^2}=\sqrt{16}=4 \neq x\)




Since \(a^2 = (+a)^2 = (−a)^2\), the numbers +a and −a are square roots of \(a^2\). If \(a ≥ 0\), then +a is the nonnegative square root of \(a^2\), and if \(a < 0\), then −a is the nonnegative square root of \(a^2\). Since \(\sqrt{a^2}\) denotes the nonnegative square root of \(a^2\), it follows that

\(\sqrt{a^2}=+a\) if \(a \geq 0\)
\(\sqrt{a^2}=-a\) if \(a < 0\)

That is, \(\sqrt{a^2}=|a|\)


PROPERTIES OF ABSOLUTE VALUE





The result in part (b) can be extended to three or more factors. More precisely, for any n real numbers, \(a_1, a_2,...,a_n\), it follows that

\(|a_1a_2 ··· a_n|=|a_1||a_2|···|a_n| \)

In the special case where \(a_1, a_2,...,a_n\) have the same value, \(a\), it follows that

\(|a^n|=|a|^n\)


GEOMETRIC INTERPRETATION OF ABSOLUTE VALUE

The notion of absolute value arises naturally in distance problems. For example, suppose that A and B are points on a number line that have coordinates \(a\) and \(b\), respectively.
Depending on the relative positions of the points, the distance \(d\) between them will be \(b − a\) or \(a − b\) (see figure below). In either case, the distance can be written as \(d = |b − a|\), so we have the following result.






This distance formula provides useful geometric interpretations of some common mathematical expressions




INEQUALITIES WITH ABSOLUTE VALUES

Inequalities of the form \(|x − a| < k\) and \(|x − a| > k\) arise so often that we have summarized the key facts about them




NOTE: The statements above remain true if < is replaced by ≤ and > by ≥, and if the open dots are replaced by closed dots in the figure above.