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Material Type: Notes; Class: College Algebra; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 1989;
Typology: Study notes
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Name: Date: Instructor:
Topics: Properties of Inequalities, Interval Notation, Types of Inequalities
I. Properties of Inequalities (pp.146 – 147) Summary of Properties:
II. Solving Linear Inequalities (p.147) A linear inequality in one variable can be written as ___________________ where a ≠ 0.
Ex. − 3 p − 2 ≤ 1
Ex. − 5 x − 3 < 7
Ways to write the solution to an inequality:
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)
Ex. { x 2 < x < 5} means that x is between 2 and 5. The interval notation for that is (2, 5).
Ex. { x 1 ≤ x ≤ 4}means that x is between 1 and 4 and includes the values of 1 and 4. The
interval notation for this is [1, 4].
Ex. { x − 2 ≤ x ≤ 0}means ______________________________________________________
The interval notation for this is ___________.
Ex. { x − 2 < x ≤ 5}is sometimes called a _____________ interval because it has one < and one
≤. The interval notation for this is _________________.
A three-part inequality (p. 149) is sometimes called a compound inequality. This type is solve by working all three parts at the same time.
Ex. − 3 ≤ 2 x − 1 < 9 Read as “2 x – 1 is between –3 and 9”
Ex.
x − − ≤ < −
Multiply all three parts by _____ to clear the fraction.
Applications of Linear Inequalities (p.149) Revenue, Cost, and Break Even Problems- Recall: “at least” translates into a “Revenue ≥ Cost” inequality. Ex. The cost to produce x units of radios is C = 20 x + 300 , while the revenue is R = 30 x. Find the interval where the product will at least break even. x = C = R =
III. Quadratic Inequalities (pp. 150 – 153)
A quadratic inequality is one that can be written as __________________________, where a, b, and c are real numbers and a ≠ 0. These inequalities also can be solved when they are <, >, or ≥ types. Recall: “ ≥^0 ” means positive and “ ≤^0 ” means negative. Steps to solving a quadratic inequality: