Section 2.7 Linear Inequalities in One Variable 177 Student Check 5 a. Darryl must make a 100 or better on his fifth test to have a test average of at least a 90. b. Rosita needs at least a 42.5 on her final exam to have a final grade of at least 70. She needs at least an 75.8 on her final exam to have a final grade of at least 80. c. Lisa can talk no more than 33 additional minutes to have a monthly bill less than or equal to $75. SUMMARY OF KEY CONCEPTS 1. The graph of the solution set of an inequality is a picture of all real numbers that make the inequality a true statement. A parenthesis (used with < or >) on a number indicates the number is not included in the solution set but every number very close to it is included. A bracket (used with ≤ or ≥) on a number indicates the number is included in the solution set. 2. Interval notation is a concise way to represent the solution set of an inequality. It represents the interval of real numbers in which solutions lie. The interval notation always begins with the left bound of the solution set and ends with the right bound of the solution set. When a set continues indefinitely to the right, the right bound is represented by ∞. When a set continues indefinitely to the left, the left bound is represented by -∞. A parenthesis is always used with ∞ or -∞. 3. Set-builder notation is another way to represent the solution set of an inequality. It is written using 5 6. We write Evariable u final inequality F. Example: Eyuy < 5F. 4. Linear inequalities are solved using the addition and multiplication properties of inequality. The most important thing to remember is that when you multiply or divide by a negative number, you must also reverse the inequality symbol. 5. Applications are solved by translating the given statements into appropriate inequalities. Key phrases are “is at least” and “is at most.” A good way to remember the translations is to think of money. For example, if you have at least $10, you would have $10 or more; if you have at most $10, you would have $10 or less. GRAPHING CALCULATOR SKILLS The calculator can be used as a reference to check the work we have done by hand. Example: x - 5 > 2 By adding 5 to both sides, we find the solution of x - 5 > 2 to be x > 7. The graph is –1 0 1 2 3 4 5 6 7 8 9 10 11 To check our work on the calculator, we should determine if the inequality is true for a number larger than 7, if it is false for a number less than 7, and what happens at 7. We will use the store feature to test values. We first store in a value for x using the STO> command. Then we enter the original inequality in the calculator and press Enter. The result will either be 1 or 0. If the result is 1, then the stored value satisfies the inequality and is a solution; if the result is 0, then the stored value does not satisfy the inequality and is not a solution. Check: x = 8 (a value larger than 7). 8 T T u n ENTER T u n 2 5 2nd MATH 3 ENTER 2 The result of 1 confirms that x = 8 is a solution of the inequality. The shaded portion of the graph should contain 8 (to the right of 7). Check: x = 6 (a value less than 7). 6 T T u n ENTER T u n 2 5 2nd MATH 3 ENTER 2 The result of 0 means that x = 6 is not a solution of the inequality. The shaded portion of the graph should not include the side with 6. Check: x = 7. T u n 2 5 2nd MATH 3 ENTER 2 T u n ENTER T The result of 0 means that x = 7 is not a solution of the inequality. The graph should not include 7, so we put a parenthesis on 7.
hendricks_beginning_algebra_1e_ch1_3
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