Since not everything in life comes in "absolutes" and not everything is either black or white, but may include shades of gray, mathematics must include words and symbols to represent relationships other than "equal".
These occasions call for use of words like "at least", "at most", "minimum", "maximum", or "between".
When negotiating for salary during a job interview an applicant might state his/her requirements as "I would like to earn at least $30,000 per year".
When shopping for a new home the buyer might declare "I cannot pay more than $100,000".
Ideal weight charts for various heights state a range considered to be within normal limits, ie. a female 5 feet 4 inches tall can weigh between 100 and 130 lbs.
Since the real world uses these vocabulary words, mathematics responds with symbols which indicate the same concepts.
Inequalities are "models" of situations with limitations described by the above phrases. Mathematically, solutions to inequality sentences are expressed symbolically and graphically with particular attention to boundary points and direction of the arrow in the graph. Example (1):
Rules for solving inequalities are basically the same as for equations except as follows: -3x/-3 > 21/-3 x > -7  Compound Inequalities are like compound English sentences...containing the words AND or OR. For an AND statement to be true, both of the limitations must be true,ie. x > -2 AND x < 6 can be written as a single statement as follow: -2 < x < 6 and graphically, looks like "between" 2 boundary points.  For an OR statement to be true, one or the other or both of the limitations must be true. Graphically, the solution usually has 2 "arms" pointing in opposite directions, ie. x < -2 OR x > 9.  Absolute Value Inequalities Absolute value denoted (| |) is defined as the distance a point is from 0 when pictured on a number line. So...absolute value is never negative. Since + 3 and -3 are each 3 units from 0, their absolute values are equal. |+3| = 3 and |-3| = 3. Correspondingly, if we are looking for a point a specific distance from 0, there are always 2 solutions, ie. |x| = 7 could have an x value of either +7 or -7. With respect to inequalities, |x| < 7 means all points x that are less than 7 units from 0.  which can be written as -7 < x < 7. Note: The graph is a great visualization of the concept "between". The above findings allow us to create a method to "split" a "less than" absolute value inequality into two separate inequalities which then can be solved separately. For example, |x| < 10 really means x is between -10 and + 10 which is written as -10 < x < 10 which can be written as -10 < x AND x < 10. A more complex example is needed here for explanation. splits into -12 < 2x - 3 AND 2x - 3 < 12 which are now solved separately using Algebra By extending this notion, the rule for splitting the "greater than" absolute value inequalities should be evident. The graph should also be visualized as the opposite of the graph pictured above and graph like an "OR". |
