MATH 21
Lesson Two

         

CHAPTER #2 - EQUATIONS, INEQUALITIES, AND PROBLEM SOLVING (pp. 44 to 104)

LESSON #2.1 - The General Objectives of this lesson is to study first-degree equations in one variable.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to:

WORD PROBLEMS - Included in the lesson.

2.1 - Solving First-Degree Equations - To begin our discussion of solving first-degree equations in one variable we need the following definitions.

  1. An equation is a mathematical statement that expresses the relation of equality between two expressions.

    a)  15 - 2 = 13

    b)  3x + 6 = 12

    c)  x2 - 4 = (x+2)(x-2)

  2. A solution is any value of the variable that satisfies the equation. For example for the equation -

    3x + 6 = 12

    the value of x = 2 yields

    3*2 + 6 = 12.

    Since this value of x made the equation a true statement, it is a solution. It should be noted that this particular equation only has one solution.

  3. An identity is an equation that is true for all values of the variable. For example the equation -

    x2 - 4 = (x+2)(x-2)

    is true for all values of x.

  4. Equivalent equations are equations that have the same solution. For instance -

    a)  3x + 6 = 12

    b)  3x = 6

    c)  x = 2

    all have the same solution, x = 2, therefore they are equivalent equations.

  5. A first-degree equation in one variable is one where the highest exponent on the variable is one. Examples include:

    a)  3x + 6 = 12

    b)  x - 17 + 2(x+1) = 3(x-1) + 5x

    First degree equations are solved by reducing them to equivalent equations, where the solution is obvious. This is done by using the following two properties of numbers.

Addition/Subtraction Property of Equality - The addition (subtraction) of the same number to (from) both sides of an equation produces an equivalent equation. That is, if

P = Q

then   P + R = Q + R

or   P - R = Q - R.

Multiplication/Division Property of Equality - The multiplication (division) of both sides of an equation by the same number produces an equivalent equation. That is, if

P = Q

then   P*R = Q*R

or   P/R = Q/R.

 

Example - Solve the equation: x -

17 + 2(x+1) = 3(x-1) + 5x.

 

Use of Equations to Solve Problems - As you can see from the previous two examples, solving first-degree equations in one variable is not too difficult. On the other hand, solving word problems, which give rise to first-degree equations, often causes us much anxiety and grief. The difficulty occurs in translating the problem expressed in standard language into mathematical terms. Fortunately, for all of us, there are some rules that can be followed which can help us to overcome this difficulty.

Rules for Solving Word Problems -

Board and Class Problem - Page 51/ 59, 65.

 

LESSON #2.2 - The General Objective of this lesson is to develop a procedure for solving fractional equations.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to: Identify a fractional equation. Solve fractional first-degree equations in one variable.

2.2 - Equations Involving Fractional Forms - Any equation containing one or more fractions can be classified as a fractional equation. Usually the first step to be carried out is to multiply both sides of the equation by the least common denominator (LCD).

Least Common Denominator - To find the LCD you follow the following procedure.

Board and Class Problem - Find the sum of

Procedure for Solving Fractional Equations -

Board and Class Problems - Pages 58 to 59/22, 29, 49.

 

LESSON #2.3 - The General Objective of this lesson is to deal with equations involving decimal coefficients.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able: Solve discount problems. Solve sales problems. Solve simple interest problems.

2.3 - Equations Involving Decimals and Problem Solving - Any equation containing decimal coefficients can be solved as previously described, however, it is often convenient to clear the equation of all decimals by multiplying by an appropriate power of ten. This often occurs with discount, sales, and investment problems.

Discount Problems - Let

P = original price
S = sales price
D = discount
then

S = P - D

Sales Problems - Let

P = profit
S = sales price
C = cost
then
S = C + P

Investment Problems - Let

P = principal invested for one year
r = interest rate
I= interest earned for one year
t = time period
then
I = rP

Board and Class Problems - Pages 65 to 66/29, 41.

 

LESSON #2.4 - The General Objective of this lesson is to deal with literal equations or formulas.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to manipulate formulas, solving for each variable in terms of the others.

2.4 - Formulas - Formulas are rules that are stated in symbolic form, usually as equations. As we saw in the word problem involving temperatures, formulas can be solved for a specific variable if we are given numerical values for the other variables. We can also manipulate the formula enabling us to solve for any variable in terms of the other, using our rules for solving equations.

Board and Class Problem - The following formula is used by electrical engineers in studying simple parallel circuits.

where R1, R2 and R3 are three parallel resistance's in the circuit, and R is the total resistance of the circuit. Solve for R in terms of the other three resistance's. Write down each step in the solution.

Formulas and Problem Solving - Solving word problems is simplified by developing common techniques that can be used to solve any problem. It also depends upon knowing the formula that applies to the particular problem being investigated. We will consider some different types now.

 

Simple Interest - The equation for simple interest for t years is given by -

A = P + Prt

where

Board and Class Problem - Page 76/49.

Uniform Motion - The equation for uniform motion is given by -

d = vt

where

Word problems of uniform motion usually involve two different motions. The following table helps organize such problems and is a useful aid for developing the appropriate equation.

Trip Distance Speed Time
Motion 1 d1 v1 t1
Motion 2 d2 v2 t2

Board and Class Problem - Page 76/55.

Mixture Problems - Many problems involve mixing solutions or substances containing different percentages of ingredients to obtain a solution or a substance containing a specified proportion of materials. These problems can be solved, by identifying some quantity that is held constant before and after the mixing process. This fact is then expressed as an equation, which can be solved. Since possibly there is more than one quantity held constant, it may be necessary to obtain more than one equation before a useful one is found.

Board and Class Problem - Page 76/61.

 

LESSON #2.5 - The General Objective of this lesson is to introduce the topic of inequalities.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve linear inequalities.

2.5 - Inequalities - Up to now our attention has been focused on equalities. We have found that given an equation such as -

2x + 3 = 8

there is only one value of x (x=2), which made it a true statement. In this lesson we are going to study inequalities such as;

2x + 3 <8
or
2x + 3 > 8.

Instead of only finding one value of x which satisfies these statements, we will find that there are many values of x which make them correct.

Definition - An inequality is a statement that an expression representing some real number is greater than (or less than) another expression representing a real number.

Symbolically - Given two real numbers x and y;

  1. x > y means that x is greater than y .
  2. x < y means that x is less than y.
  3. x > y means that x is greater than or equal to y.
  4. x < y means that x is less than or equal to y.

Geometrically -

  1. Statement #1 means that x is to the right of y.
  2. Statement #2 means that x is to the left of y.
  3. Statement #3 means that x includes or is to the right of y.
  4. Statement #4 means that x includes or is to the left of y.

Just as we reduce equations to equivalent ones where the solution is obvious, inequalities can be solved the same way. However, care must be exercised when multiplying inequalities. The following properties are used to achieve this result.

Addition/Subtraction Property of Inequality - For all real numbers P, Q and R, the addition(subtraction) of the same number to(from) both sides of an inequality produces an equivalent inequality. That is, if

P > Q
then
P + R > Q + R
or
P - R > Q - R.

Multiplication/Division Property of Inequality - For all real numbers P, Q and R,

  1. If R > 0 , then the multiplication (division) of both sides of an inequality by the same number produces an equivalent inequality. That is, if

    P > Q
    then
    P*R >Q*R
    or
    P/R > Q/R.

  2. If R <0 , then the multiplication(division) of both sides of an inequality by the same number produces an equivalent inequality where the inequality sign has been reversed. That is, if 

    P> Q
    then
    P*R < Q*R or P/R < Q/R.

Interval Notation - Discuss the notation used in the text to indicate bounded and unbounded intervals.

Board and Class Problem - Solve the inequality -

x -17 + 2(x+1)> 3(x-1) + 5x.

As you can see the technique for solving linear equations and linear inequalities is basically the same. The one thing you have to remember is to reverse the inequality sign if you multiply or divide by a negative number.

 

LESSON #2.6 - The General Objective of this lesson is lesson is to apply inequalities to problem solving.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve problems involving linear inequalities.

2.6 - More on Inequalities and Problem Solving- The rules for handling fractions and decimals in inequalities are basically the same as those for equations. Again, you have to remember to reverse the inequality sign if you multiply or divide by a negative number.

Compound Statements - The words and and or are used to form compound statements.

Conjunctions - Compound numerical statements that use and are called conjunctions. Conjunctions are true only if all of its component parts are true.

Disjunctions - Compound numerical statements that use or are called Disjunctions. Disjunctions are true if at least one of its component parts is true.

This leads to two concepts in set theory.

Problem Solving - It is important to understand inequalities, because in the real world many problems are posed in terms of being greater than something or less than something. Consider the following problems.

Board and Class Problems - Page 92/57, 60.

 

LESSON #2.7 - The General Objective of this lesson is lesson is to study absolute value equations and inequalities.

PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve problems involving -

2.7 - Equations and Inequalities Involving Absolute Value - In Section 1.2 the absolute value of a number was defined as:

Absolute Value - The absolute value of any real number x is defined as;

  1. |x| =  x ,   if  x  > 0.
  2. |x| = -x ,  if x  <  0.

Let us apply this definition to absolute value equations and inequalities.

HOMEWORK - Lesson Two Assignment.

COMMENTS

 

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