Algebra 2 Part6 Chapter 1

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This section is called "Solving Linear Inequalities in One Variable". "A 'linear inequality' is a relationship involving <, < _,>, or >_ instead of =. 'Linear inequalities can be solved the same way linear equations are, but with one important modification to the multiplication property. In an inequality, if both sides are multiplied (or divided) by the same "negative' quantity, the direction of the inequality changes. This is illustrated in the next example. Example 1.9 Solve -3x + 7 < 25 for x. Solution -3x + 7 < 25 -7 <-7 ----------------- -3x/-3<18/-3 x>-6 "Note that because we divided both sides by -3, the direction of the inequality changed from < to>. This quirk of inequalities can be avoided by aways moving the variable to the side where its coefficient will be positive. However, this can introduce a second difference between equations and inequalities. If the two sides of an inequality are switched right to left, the direction of the inequality must be reversed. For example, to rewrite 3 > x with the variable on the left, you must switch the direction of the inequality to get x < 3." Example 1.10 Solve 3(2x + 4)> _ 10x - (x-6) for x. solution 3(2x + 4) >_ 10x-(x -6) 6x +12 >_ 10x - x + 6 6x + 12 >_ 9x + 6 -6x - 6 >_-6x - 6 6/3 >_ 3x/3 2 >_ x or x<_2 "When dividing by a positive quantity, the direction of an inequality does not change. When reversing the order of the two sides of an inequality, the direction changes." Interval Notation "'Inequalities in one variable represent an 'interval' on the 'number line'. 'Interval notation' is a compact way to write inequalities. Here is what you need to know: 1. Intervals are always written left to right, smaller number to larger number. 2. [a, b] means {x | a<_ x <_ b} This is called a 'closed interval'; the endpoints are included 3. (a, b) means {x | a < x < b}. This is called an 'open interval'; the endpoints are 'not' included. You need to use 'context' to avoid confusing it with an ordered pair. 4. [a, b) and (a, b] are called 'half-open intervals'. [a, b) means {x | a <_ x < b}, while (a,b] means {x | a < x <_ b}. 5. 'Unbounded intervals' are written using eternity circles and/or -eternity; eternity and -eternity a. are not included in the interval. For example, for x>_5,, we write 5, eternity). 6. In 'interval notation', U(union) replaces v (or) and upside down U(intersection) replaces ^ (and) as notation for writing two or more intervals.