Unit 3: Equations, Inequalities, and Variations

The absolute value of a number is defined as its distance from zero on a number line. Written as a value between two vertical lines, there are several basic properties of absolute values:

• |a| = |-a|
• |a - b| = |b - a|
• |a| * |b| = |ab|
• |a| / |b| = |a/b|

Solving equations with absolute values is very similar to solving any other equations, except that absolute values yield two possibilities. In order to remove the absolute value sign, you must set up two equations; one for the value inside the absolute value sign, and one for its negative. Hence, |a| = 5 will yield two answers: 5 and -5. However, always check for extraneous solutions by plugging both answers back into the original equation (especially the negative solution).

Absolute values in inequalities is also a relatively simple concept. Also take the value within the absolute value symbol and its negative, and create two equations for two solutions. If |b+2| < 6, then b+2 < 6, making b < 4, and -b - 2 < 6, making b > -8.

Quadratic inequalities are more complicated than normal inequalities. First off, the quadratic should be factored.
Next, the critical values (values for x at which the quadratic equals 0) should be found.
Several ranges, written in interval notation, should be written from negative infinity to the lowest critical value, then from that value to the next critical value, and so on until you reach the highest value to infinity. Whether or not the critical values should be included in the intervals depends on whether the quadratic is greater/less than and/or equal to.
Using a chart, you need to determine whether any given value in an interval will yield a negative or a positive result. This should be done for all intervals. The intervals that corrispond to your original inequality will represent the range(s) of your solutions.

Rational expressions can usually be simplified by cancelling out like terms. It is also necessary to determine the domain by finding values that make the expression invalid, and disregarding these values in your solution.

Rational equations involve two rational equations that are equivalent to each other. Simply cross multiply or find the Least Common Denominator (LCM) and then solve as you would a normal equation. However, watch out for invalid solutions by discluding any impossible values (i.e. negatives under a square root).

Rational inequalities are similar to quadratic inequalities in that they also involve critical values. In order to determine the critical values, both the numerator and denominator of hte rational expression must be factored, and each of these factors must be set to zero. From there, you can solve for your variable as you would in a quadratic inequality. Always remember to state your domain restrictions.

Direct variation refers to a proportion between an x value and a y value, such that x/y always equals a constant.

Inverse variaiton is similar to direct variation, except that xy always equals a constant, so x decreases as y increases.