Solving+Inequalities

Rule 1. Adding/subtracting the same number on both sides. **Example:** The inequality //x//-2>5 has the same solutions as the inequality //x// > 7. (The second inequality was obtained from the first one by adding 2 on both sides.) Rule 2. Switching sides and changing the orientation of the inequality sign. **Example:** The inequality 5-//x//> 4 has the same solutions as the inequality 4 < 5 - //x//. (We have switched sides and turned the ``> into a ``<). Last, but not least, the operation which is at the source of all the trouble with inequalities: Rule 3a. Multiplying/dividing by the same POSITIVE number on both sides. Rule 3b. Multiplying/dividing by the same NEGATIVE number on both sides AND changing the orientation of the inequality sign. **Examples:** This sounds harmless enough. The inequality has the same solutions as the inequality. (We divided by +2 on both sides). The inequality -2//x// > 4 has the same solutions as the inequality //x//< -2. (We divided by (-2) on both sides and switched "> to "<.) But Rule 3 prohibits fancier moves: The inequality DOES NOT have the same solutions as the inequality //x// > 1. (We were planning on dividing both sides by //x//, but we can't, because we do not know at this point whether //x// will be positive or negative!) In fact, it is easy to check that //x// = -2 solves the first inequality, but does not solve the second inequality. www.googel.com/inequalities

__Solving Compound Inequalities__

Examples of conjunctions: x > -5 and x <1 y -3
 * Compound inequalities are two inequalities considered together
 * A compound inequality containing the word and is true only if both inequalities are true. This type of compound inequality is called a **conjunction**.

**The absolute value of a number measures its distance to the origin on the real number line.** Since 5 is at 5 units distance from the origin 0, the absolute value of 5 is 5, |5|=5 ||
 * [[image:http://www.sosmath.com/algebra/inequalities/pictures/pic10.gif width="500" height="119" align="center"]]