Published On:Thursday, 8 December 2011
Posted by Muhammad Atif Saeed
Inequalities, Graphing, and Absolute Value
Inequalities
An inequality is a statement in which the relationships are not equal. Instead of using an equal sign (=) as in an equation, these symbols are used: > (is greater than) and < (is less than) or ≥ (is greater than or equal to) and ≤ (is less than or equal to).Axioms and properties of inequalities
For all real numbers a, b, and c, the following are some basic rules for using the inequality signs.- Trichotomy axiom: a > b, a = b, or a < b.
These are the only possible relationships between two numbers. Either the first number is greater than the second, the numbers are equal, or the first number is less than the second. - Transitive axiom: If a > b, and b > c, then a > c.
Therefore, if 3 > 2 and 2 > 1, then 3 > 1.If a < b and b < c, then a < c.
Therefore, if 4 < 5 and 5 < 6, then 4 < 6. - Addition property:
Adding or subtracting the same amount from each side of an inequality keeps the direction of the inequality the same.- If a > b, then a + c > b + c.
- If a > b, then a – c > b – c.
- If a < b, then a + c < b + c.
- If a < b, then a – c < b – c.
Example: If 3 > 2, then 3 + 1 > 2 + 1 (4 > 3) If 12 < 15, then 12 – 4 < 15 – 4 (8 < 11) - Positive multiplication and division property:
Multiplying or dividing each side of an inequality by a positive number keeps the direction of the inequality the same.- If a > b, and c > 0, then ac > bc.
- If a < b, and c > 0, then ac < bc.
- If a > b, and c > 0, then
.
- If a < b, and c > 0, then
.
Example: If 5 > 2, then 5(3) > 2(3), therefore, 15 > 6. If 3 < 12, then.
- Negative multiplication and division property:
Multiplying or dividing each side of an inequality by a negative number reverses the direction of the inequality.- If a > b, and c < 0, then ac < bc.
- If a < b, and c < 0, then ac > bc.
- If a > b, and c < 0, then
.
- If a < b, and c < 0, then
.
Example: If 5 > 2, then 5(– 3) < 2(– 3); therefore, – 15 < – 6. If 3 < 12, then.
Solving inequalities
When working with inequalities, treat them exactly like equations (except, if you multiply or divide each side of the inequality by a negative number, you must reverse the direction of the inequality).Example 1
Solve for x: 2 x + 4 > 6.
Example 2
Solve for x: –7 x > 14.Divide by –7 and reverse the direction of the inequality.
Example 3
Solve for x: 3 x + 2 ≥ 5 x – 10.
Notice that opposite operations are used. Divide each side of the inequality by –2 and reverse the direction of the inequality.
In set builder notation, { x: x ≤ 6}.
Graphing on a Number Line
Integers and real numbers can be represented on a number line. The point on this line associated with each number is called the graph of the number. Notice that number lines are spaced equally, or proportionately (see Figure 1).Figure 1. Number lines.
Graphing inequalities
When graphing inequalities involving only integers, dots are used.Example 1
Graph the set of x such that 1 ≤ x ≤ 4 and x is an integer (see Figure 2).{ x:1 ≤ x ≤ 4, x is an integer}
Figure 2. A graph of {x:1 ≤ x ≤ 4, x is an integer}.
Example 2
Graph as indicated (see Figure 3).- Graph the set of x such that x ≥ 1.
{ x: x ≥ 1} - Graph the set of x such that x > 1 (see Figure 4).
{ x: x > 1} - Graph the set of x such that x < 4 (see Figure 5).
{ x: x < 4}
Figure 3. A graph of { x: x ≥ 1}.
Figure 4. A graph of { x: x > 1}
Figure 5. A graph of { x: x < 4}
Intervals
An interval consists of all the numbers that lie within two certain boundaries. If the two boundaries, or fixed numbers, are included, then the interval is called a closed interval. If the fixed numbers are not included, then the interval is called an open interval.Example 3
Graph.- Closed interval (see Figure 6).
{ x: –1 ≤ x ≤ 2} - Open interval (see Figure 7).
{ x: –2 < x < 2}
Figure 6. A graph showing closed interval { x: –1 ≤ x ≤ 2}.
Figure 7. A graph showing open interval { x: –2 < x < 2}.
Example 4
Graph the half-open interval (see Figure 8).{ x: –1 < x ≤ 2}
Figure 8. A graph showing half-open interval { x: –1 < x ≤ 2}.
Absolute Value
The numerical value when direction or sign is not considered is called the absolute value. The absolute value of x is written | x|. The absolute value of a number is always positive except when the number is 0.| 0 | = 0| x | > 0 when x ≠ 0 | – x| > 0 when x ≠ 0
Example 1
Give the value.- |4| = 4
- |–6| = 6
- |7 – 9| = |–2| = 2
- 3 – |–6| = 3 – 6 = –3
