86 Chapter 2 Linear Equations and Inequalities in One Variable Some examples of linear inequalities in one variable are x >3 2x + 3≤-7 7x - 2 ≥ 6x +4 4(3y - 1)<-2(y - 9) A solution of a linear inequality is a value that makes the inequality true. The solution set of an inequality is the set of all the solutions. Consider the inequality x > 3. Some solutions of this inequality are 3.05, 3.9, 4, 4.2, 10, and 250. There are infinitely many solutions of this inequality since there are infinitely many numbers larger than 3. It is impossible to list all of the solutions, so we visualize the solution set by graphing the solution set on a number line. We also use a special notation to represent the solution set. The solution set of x > 3 includes all the numbers greater than 3 but not equal to 3. So, the graph of the solution set of x > 3 looks like the following. 6 t f t m t t c t t –2 –1 0 1 2 3 4 5 7 8 Notice that a parenthesis is used when the endpoint is not included in the solution set. A bracket is used when the endpoint is included in the solution set. We can also write the solution set of an inequality using two special notations, interval notation and set-builder notation. Interval notation is a concise way to represent the solution set of an inequality. In interval notation, we represent the smallest and largest values in the solution set. • When the graph of an inequality extends to the right indefinitely, we say that the numbers in the set approach ∞ (infinity). This means that the numbers in the solution set of the inequality increase indefinitely. • When the graph of an inequality extends to the left indefinitely, we say that the numbers in the set approach -∞ (negative infinity). This means that the numbers in the solution set of the inequality decrease indefinitely. • Interval notation makes use of parentheses and brackets, like graphing solution sets of inequalities. • A bracket, or , denotes that the endpoint of the solution set is included. • A parenthesis, ( or ), denotes that the endpoint of the solution set is not included. • Parentheses are always used on positive and negative infinity. So, the solution set of x > 3, written in interval notation, is (3, ∞). 6 ftm t 3 tm t ∞ –2 –1 0 1 2 3 4 5 7 8 Set-builder notation was discussed in Chapter 1. Recall that set-builder notation is used to state the conditions the solutions must meet to be included in the set and is written in the form 5x ucondition x must satisfy6. So, the solution set of x > 3, written in set-builder notation, is 5xux > 36. Mathematical Notation Verbal Expression (3, ∞) The interval from 3 to infinity, not including 3 5xux > 36 The set of x such that x is greater than 3
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