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Section 2.7 Linear Inequalities in One Variable 167 1b. To graph the solution of x ≤ 4, we shade the portion of the number line that corresponds to values less than 4 or equal to 4. Numbers less than 4 are numbers to the left of 4. A bracket is used to show that 4 is included in the solution set since 4 equals 4. –2 –1 0 1 2 3 4 5 6 7 8 1c. The inequality 1 2 < x means that 1 2 is smaller than all of the solutions, or that all of the solutions are larger than 1 2 . So, 1 2 < x is equivalent to x > 1 2 . Therefore, we shade the portion of the number line to the right of 1 2 and place a parenthesis on 1 2 since it is not included in the solution set. –5 –4 –3 –2 –1 0 1 2 3 4 5 ���������� The inequality c < x is equivalent to x > c. 1d. To graph the solutions of -2 < x ≤ 5, we find the values that are between -2 and 5, including 5. So, we shade the portion of the number line that lies between these endpoints. The number -2 is not included since -2 is not less than -2. –5 –4 –3 –2 –1 0 1 2 3 4 5 Student Check 1 Graph the solution set of each inequality on a number line. a. x > -2 b. x ≤ -4.2 c. 5 3 > x d. -4 ≤ x < 2 Examples 1a–1c illustrate a simple inequality. The solutions of simple inequalities have to satisfy one only inequality. Example 1d illustrates a compound inequality. The solutions of compound inequalities must satisfy two inequalities, not just one. For instance, -2 < x ≤ 5 means that “-2 < x and x ≤ 5” or “x > -2 and x ≤ 5” Interval Notation Solutions of inequalities can also be represented in a concise way that represents the smallest and largest values in the solution set. This notation is called interval notation. 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 continue getting larger without bound. 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 continue getting smaller without bound. Interval notation makes use of parentheses and brackets like graphing solution sets of inequalities. If the endpoint is included in the solution set of the inequality, then a bracket is used with the endpoint. If the endpoint is not included, then a parenthesis is used. Objective 2 ▶ Write the solution set of an inequality in interval notation.


hendricks_beginning_algebra_1e_ch1_3
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