Math 55

Lesson 1 Overview

1.1 Solving Equations

First of all, let’s discuss the difference between an expression and an equation.

Examples (exs) of expressions: 2x, 4y-5z, 3x+2a-2+a

Exs of equations: 2x=10, x+2=5, 3y+2-y=4+y

Did you notice that the main difference is that equations have an equals sign, but expressions do not? (To help you remember, equations and equals both start with "eq"). Expressions can be simplified, but not solved. The common directions for an expression is to "simplify the expression," and the answer is usually a combination of numbers and variables. On the other hand, we can solve equations to see for which value(s) of the variable makes the math sentence true. The answer, called the solution, is usually a number.

To determine if a certain number is a solution of an equation, just plug that value in for the variable and see if it makes a true statement!

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Ok, so now we’re ready to learn how to solve equations!

Goal: Solve for x. (That means isolate x, or get x alone on one side of the equals sign.) We will do this using inverse operations. That is, to get rid of subtraction, we will add. To get rid of division, we will multiply. Etcetera.

Method:

Combine like terms on the same side of the equals sign, if any.

Take all the variable terms (terms that contain the variable) to one side of the equals sign and take all the constant terms to the other side. (In the following examples, I will show variable terms in blue and constant terms in black.)

Ex: Solve t - 3=19

There is only one variable term, so take the constant term that’s next to it to the other side. Since 3 is being subtracted from t, we will want to add 3 to both sides to get rid of -3 on the Right Hand Side (RHS):

t – 3 +3 = 19 +3

t + 0 = 22 Simplify Left Hand Side (LHS) (Anything + 0 = itself)

t = 22

t - 3=19 Original problem

22 - 3=19 ? Plug in 22 for t and simplify the LHS by combining 22 and -3.

Ex Solve -8 + y = 17

There is only one variable term, so take the constant term that’s next to it to the other side. To get rid of -8 from the LHS, we will add 8 to both sides:

+8 -8 + y = +8 +17

0 + y = 25 Simplify LHS.

y = 25 The solution.

Ex Solve -6x = 108

Since there are no like terms on the same side of the =, and the variable term is on one side and the constant term is on the other side, let’s see what’s bothering x. Since -6 is multiplying the x, so to get rid of it, we need to divide both sides by (-6):

-6x/(-6) = 108/(-6)

1x = -18 (since any non-zero number divided by itself equals 1)

x = -18 (since 1x= 1*x = x)

Since 9 is dividing the –y, multiply both sides by 9, resulting in –y = 126 which is the same as -1y = 126. At this point, you can either multiply both sides by -1 (remember that just changes the signs), or you can think of it as:

-1y = 126 means -1 is multiplying y, so we can divide both sides by (-1):

-1y/(-1) = 126/(-1)

y = -126 which is the solution.

Ex Solve 2x – 2 = -3x + 3

We do have variable terms on both sides and constant terms on both sides. Let’s first pick a side to take the variable terms, say the LHS. To "move" -3x to the other side, we need to +3x to both sides:

+3x +2x – 2 = +3x -3x + 3

5x – 2 = 3 Next +2 to both sides to get the constants on the RHS.

5x – 2 +2 = 3 +2

5x = 1 Now divide both sides by 5 to get x alone.

5x /5= 1/5

x = 1/5

1.4 Sets, Inequalities, and Interval Notation

1.5 Intersections, Unions, and Compound Inequalities

The intersection of two sets A and B is the set of all members that are common to both set A and set B. It is denoted by A∩ B

The union of two sets A and B is the set of all members that are in set A and all members that are in set B. It is denoted by A U B. (Do not write repeats twice.

Empty set is the intersection of two sets that have nothing in common. It is denoted by A ∩ B = Ø

Ex

Find the intersection of {2,3,4,6,8} and {1,4,6,7,9}

{2,3,4,6,8} ∩ {1,4,6,7,9} = {4,6}

Ex

Find the union of {2,3,4,6,8} and {1,4,6,7,9}

{2,3,4,6,8} U {1,4,6,7,9} = {1,2,3,4,6,7,8,9}

Ex

The conjunction (intersection) of inequalities: -3 < x and x < 2

Can be expressed as -3 < x < 2

Or (-3, ∞) ∩ (-∞ , 2) (Unsimplified)

Or (-3, 2) (Simplified) all values between -3 and 2, not including -3 and 2

Ex

The disjunction (union) of inequalities: 8x – 2 ≥ 6 or 5 – 3x ≤ -1

Can be expressed as

8x ≥ 8 or 6 ≤ 3x

x ≥ 1 or 2 ≤ x (Note 2 ≤ x is the same as x ≥ 2)

[1, ∞) U [2, ∞)

which can be simplified to [1, ∞)

Ex:

Water evaporates at temperatures over 100 degrees Celsius. Over what temperature does water evaporate on the Fahrenheit scale?

F = 1.8C + 32

Solve for C:

C = (F – 32)/1.8

Want to calculate C > 100, so replace C by (F – 32)/1.8:

C > 100

(F – 32)/1.8 > 100

F – 32 > 180

F > 212, Water evaporates at over 212 degrees Fahrenheit

1.6 Absolute Value Equations and Inequalities

Absolute value is defined as the distance from zero on a number line.

If you have |a| = |b|, then either they are the same value or they are the opposite of each other, i.e. a = b or a = -b

Tips for inequalities containing absolute values:

|x| < # is the same as –# < x < #

|x| ≤ # is the same as –# ≤ x ≤ #

|x| > # is the same as x < -# or x > #

|x| ≥ # is the same as x - ≤ # or x ≥ #

Ex

|x| = 2, then x = 2 or x = -2

Ex

|x + 3| = 4, then x + 3 = 4 or x + 3 = -4

x = 1 or x = -7

Ex

|x – 6| = |2x + 5|, then

x – 6 = 2x + 5, or x – 6 = -(2x + 5)

x = -11 or x = 1/3

Ex

|6x – 3| < 6, this is the same as

-6 < 6x – 3 < 6

-3 < 6x < 9

-½ < x < 3/2

Using interval notation: (-1/2, 3/2)

Quiz available only in  My Math Lab:

Due by lesson one due date on main page of course.

Exercise Sets and Discussion Questions will be given in the Discussion board.