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Notes for 1.7 Inequalities (pp. 146 – 155)
Topics: Properties of Inequalities, Interval Notation, Types of
Inequalities
I. Properties of Inequalities (pp.146 – 147)
Summary of Properties:
1. If a < b, then a + c < b + c. (Addition Property of Inequalities)
2. If c > 0 (is positive), then a < b and ac < bc. (Multiplication Property of Inequalities)
3. If c < 0 (is negative), then a < b and ac > bc are equivalent. (Switching the direction of the
inequality has to be done when multiplying or dividing by a negative value)
II. Solving Linear Inequalities (p.147)
A linear inequality in one variable can be written as ___________________ where 0a
≠
.
Ex. 321p−−≤
Ex. 537x−−<
Ways to write the solution to an inequality:
1.
2.
3. Interval notation = description of the number line graph. Uses numbers or symbols of
∞(infinity to the positive side of the number line) or
−
∞(negative infinity to the left side of the
number line), along with a parentheses or a bracket, to concisely describe the entire answer.
Rules for Interval Notation: (chart on p. 148)
1. Find “critical value(s)” of the inequality, which is/are the solutions to the companion equality.
2. Graph or imagine them on the number line.
3. Place the left value on the left, then a comma, then the rightmost value.
4.a. Use a ( or ) beside the value if the symbol beside the value is < or >. Similar to the open
circle at the point when you draw the number line.
b. Use a [ or ] beside the value if the symbol beside the value is
≤
or ≥. Similar to the shaded
circle at the point when you draw the number line.
5. This notation is a description of the number line solution, without the number line.
6. If the number line was shaded over the arrows, then you’ll use either of the infinities,
depending on the direction. You always need a ( or ) beside those, because you never close off
infinity.
7. The interval notation description of the entire number line (“all real numbers”) is
(
)
,
−
∞∞ .
Ex. {2 5}xx<< means that x is between 2 and 5. The interval notation for that is (2, 5).
