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Lecture Notes on Inequalities - College Algebra | MATH 1130, Study notes of Algebra

Material Type: Notes; Class: College Algebra; Subject: Mathematics; University: Pellissippi State Technical Community College; Term: Unknown 1989;

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

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Name:
Date:
Instructor:
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).
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Name: Date: Instructor:

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 a ≠ 0.

Ex. − 3 p − 2 ≤ 1

Ex. − 5 x − 3 < 7

Ways to write the solution to an inequality:

  1. 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.
  4. This notation is a description of the number line solution, without the number line.
  5. 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. { 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:

  1. Solve the companion quadratic equation to get critical values.
  2. Identify the intervals determined by the solutions of the equation by placing the critical values on the number line and having them act as “0”s.
  3. Use the signs (- to the left and + to the right) to represent the number line signs above the line.
  4. Determine the signs of the products of each region and write that result below the number line.
  5. The solution is the region(s) that have the desired final sign pattern.