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C ollege A lgebra

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College Algebra

Basic Algebraic Operations

(Appendix A)

Objectives

Cover the topics in Appendix A (A6 - A 8):

A-6 : Rational Exponents.

A-8 : Linear equations and inequalities.

Appendix A

A-6 : Rational Exponents

A-6 : Learning Objectives

- After completing this section, you should be able to:
- 1. Rewrite a rational exponent in radical notation.
- 2. Simplify an expression that contains a rational exponent.
- 3.Use rational exponents to simplify a radical expression.

A-6

A-6 : Rational Exponents

Rational Exponents and Roots

If x is positive, p and q are integers and q is positive,

In other words, when you have a rational exponent, the denominator of that exponent is your index or root number and the numerator of the exponent is the exponential part. I have found it easier to think of it in two parts.Â

Find the root part first and then take it to the exponential part if possible.Â It makes the numbers a lot easier to work with.

Radical exponents follow the exact same exponent rules as discussed in Integer Exponents. In that section we only dealt with integers, but you can extend those rules to rational exponents.Â

A-6

A-6 : Rational Exponents

Review of Exponential Rules

A-6

A-6 : Rational Exponents

Example 1: EvaluateÂ

*Rewrite exponent 1/2 as a square root

We are looking for the square root of 49 raised to the 1 power, which is the same as just saying the square root of 49.Â .

If your exponentâ€™s numerator is 1, you are basically just looking for the root (the denominatorâ€™s exponent).Â

Our answer is 7 since the square root of 49 is 7.

A-6

A-6 : Rational Exponents

Example 2: EvaluateÂ

*Rewrite exponent 2/3 as a cube root being squared

*Cube root of -125 = -5

In this problem we are looking for the cube root of -125 squared.Â Again, I think it is easier to do the root part first if possible.

The numbers will be easier to work with.Â The cube root of -125 is -5 and (-5) squared is 25.

A-6

A-6 : Rational Exponents

Example 3: EvaluateÂ

*Rewrite as recip. of base raised to pos. exp. *DO NOT take the reciprocal of the exponent, only the base

*Rewrite exponent 3/2 as a square root being cubed

*Square root of 49/36 = 7/6

A-6

A-6 : Rational Exponents

In this problem we have a negative exponent to start with.Â

That means we need to take the reciprocal of the base.

Note that we DO NOT take the reciprocal of the exponent, only the base.

From there we are looking for the square root of 49/36 cubed.Â

Again, I think it is easier to do the root part first if possible.Â The numbers will be easier to work with.Â

The square root of 49/36 is 7/6 and 7/6 cubed is 343/216.

A-6

A-6 : Rational Exponents

Example 4: SimplifyÂ

* Multiply like bases, add. Exp

A-6

A-6 : Rational Exponents

Example 5: SimplifyÂ

*Raise a base to two exponents, mult. exp.

*Rewrite as recip. of base raised to pos. exp.

*Cube root of 8 = 2

A-6

A-6 : Rational Exponents

Example 6: SimplifyÂ

* Divide like bases, sub. exp

A-6

A-6 : Rational Exponents

Example 7: Simplify Â by reducing the index of the radical.

*Rewrite tenth root of x squared as x to the 2/10 power

*Simplify exponent

*Rewrite exponent 1/5 as a fifth root

A-6

A-8 : Linear equations and inequalities

1 : Equations.

2 : Solving linear equations.

3 : Inequalities relations and interval notations.

4: Solving linear Inequalities

Appendix A

1.Equations

Definition:

An algebraic equation is a mathematical statement that relates

two algebraic expressions involving at least one variable.

Examples:

3x-2=7 2x2-3x+5=0

Solution Set (roots):

The solution set of an equation is defined to be the set of

elements that make the equation true.

A-8

2 : Solving linear equations.

Learning Objectives:

- After completing this section, you should be able to:
- Know what a linear equation is.
- Know if a value is a solution or not.
- Use the addition, subtraction, multiplication, and division properties of equalities to solve linear equations.
- Classify an equation as an identity, conditional or inconsistent.

A-8

2 : Solving linear equations.

Equation

Two expressions set equal to each other

Linear Equation

An equation that can be written in the formÂ ax + b = 0where a and b are constants

The following is an example of a linear equation:

A-8

2 : Solving linear equations.

Solution

A value, such that, when you replace the variable with it,

it makes the equation true.

(the left side comes out equal to the right side)

Solution Set

Set of all solutions

Solving a Linear EquationÂ in General Get the variable you are solving for alone on one side and everything else on the other side using INVERSE operations.

A-8

2 : Solving linear equations.

Addition and Subtraction Properties of Equality

If a = b, then a + c = b + c

If a = b, then a - c = b - c

Example 1:Â SolveÂ for x.

Note that if you put 8 back in for x in the original problem you will see that 8 is the solution to our problem.

A-8

2 : Solving linear equations.

Multiplication and Division Properties of Equality

If a = b, then a(c) = b(c)

If a = b, then a/c = b/c where c is not equal to 0.

Example 2:Â SolveÂ for x.

If you put -21 back in for x in the original problem, you will see that -21 is the solution we are looking for.

A-8

2 : Solving linear equations.

Strategy for Solving a Linear Equation

- Step 1: Simplify each side, if needed.
- This would involve things like removing ( ), removing fractions, adding like terms, etc.Â To remove ( ):Â Just use the distributive property.
- To remove fractions: Since fractions are another way to write division, and the inverse of divide is to multiply, you remove fractions by multiplying both sides by the LCD of all of your fractions.

- Step 2: Use Add./Sub. Properties to move the variable term to one side and all other terms to the other side.Â
- Step 3: Use Mult./Div. Properties to remove any values that are in front of the variable.
- Step 4:Â Check your answer.

A-8

2 : Solving linear equations.

Example 3:Â SolveÂ for y.

If you put -1 back in for y in the original problem you will see that -1 is the solution we are looking for.

A-8

2 : Solving linear equations.

Example 4:Â SolveÂ for x.

If you put -6 back in for x in the original problem you will see that -6 is the solution we are looking for.

A-8

2 : Solving linear equations.

Example 5:Â Solve Â for x.

If you put -11/5 back in for x in the original problem you will see that -11/5 is the solution we are looking for.

A-8

2 : Solving linear equations.

Identity

An equation is classified as an identity when it is true for ALL real numbers for which both sides of the equation are defined.

Example 6:Â Solve Â for x.

Where did our variable, x, go???Â It disappeared on us.Â Also note how we ended up with a TRUE statement, 14 does indeed equal 14.Â This does not mean that x = 14.Â

A-8

2 : Solving linear equations.

Conditional Equation

A conditional equation is an equation that is not an identity, but has at least one real number solution.

Example 7:Â Solve Â for x.

If you put 4 back in for x in the original problem you will see that 4 is the solution we are looking for. This would be an example of a conditional equation, because we came up with one solution.

A-8

2 : Solving linear equations.

Inconsistent Equation

An inconsistent equation is an equation with one variable that has no solution.

Example 8:Â Solve Â for x.

Example 8:Â Solve Â for x.

Where did our variable, x, go???Â It disappeared on us.Â Also note how we ended up with a FALSE statement, -2 is not equal to 5.Â This does not mean that x = -2 or x = 5.Â

Whenever your variable drops out AND you end up with a FALSE statement, then after all of your hard work, there is NO SOLUTION.

So, the answer is no solution which means this is an inconsistent equation.

Where did our variable, x, go???Â It disappeared on us.Â Also note how we ended up with a FALSE statement, -2 is not equal to 5.Â This does not mean that x = -2 or x = 5.Â

Whenever your variable drops out AND you end up with a FALSE statement, then after all of your hard work, there is NO SOLUTION.

So, the answer is no solution which means this is an inconsistent equation.

A-8

3 : Inequalities relations and interval notations.

Learning Objectives

- After completing this section, you should be able to:
- Use the addition, subtraction, multiplication, and division properties ofÂ inequalities to solve linear inequalities.
- Solve linear inequalities involving absolute values.
- Write the answer to an inequality using interval notation.
- Draw a graph to give a visual answer to an inequality problem.

A-8

3 : Inequalities relations and interval notations.

Inequality

An inequality says that one expression is greater than, greater than or equal to, less than, or less than or equal to, another.

Inequality Signs

Read left to right: a < bÂ Â Â a is less than b a < bÂ Â a is less than or equal to b

a > bÂ Â Â Â a is greater than b a > bÂ Â Â a is greater than or equal to b

A-8

3 : Inequalities relations and interval notations.

Interval Notation

Interval notation is a way to notate the range of values that would make an inequality true.Â There are two types of intervals, open and closed (described below), each with a specific way to notate it so we can tell the difference between the two.Â Note that in the interval notations (found below), you will see the symbolÂ , which means infinity.Â

Positive infinity () means it goes on and on indefinitely to the right of the number - there is no endpoint on the right.Â Negative infinity (-) means it goes on and on indefinitely to the left of the number - there is no endpoint to the left.Â

Since we donâ€™t know what the largest or smallest numbers are, we need to use infinity or negative infinity to indicate there is no endpoint in one direction or the other.

In general, when using interval notation, you always put the smaller value of the interval first (on the left side), put a comma between the two ends, then put the larger value of the interval (on the right side).Â You will either use a curved end ( or ) or a boxed end [ orÂ ], depending on the type of interval .

A-8

3 : Inequalities relations and interval notations.

Open Interval

An open interval does not include where your variable is equal to the endpoint.

A-8

3 : Inequalities relations and interval notations.

Closed Interval

A closed interval includes where your variable is equal to the endpoint.

A-8

3 : Inequalities relations and interval notations.

Combining Open and Closed Intervals

Sometimes one end of your interval is open and the other end is closed.Â You still follow the basic ideas described above.Â The closed end will have a [Â Â or ] on itâ€™s end and the open end will have a ( or ) on its end.Â

A-8

3 : Inequalities relations and interval notations.

Addition/Subtraction Property for Inequalities

If a < b, then a + c < b + c

If a < b, then a - c < b â€“ c

In other words, adding or subtracting the same expression to both sides of an inequality does not change the inequality.

Example 1:Â Solve, write your answer in interval notation and graph the solution set

*Inv. of sub. 10 is add. 10 *Open interval indicating all values less than 5

*Visual showing all numbers less than 5 on the number line

Interval notation:

Graph:

A-8

3 : Inequalities relations and interval notations.

Example 2:Â Solve, write your answer in interval notation and graph the solution set

*Inv. of add 4 is sub. 4 Â *Closed interval indicating all values greater than or = -3

*Visual showing all numbers greater than or = to -3 on the number line.Â

[-3, Â )

Interval notation:

Graph:

A-8

3 : Inequalities relations and interval notations.

Multiplication/Division Properties for Inequalities

when multiplying/dividing by a positive value

If a < bÂ ANDÂ c is positive, thenÂ Â ac < bc

If a < bÂ ANDÂ c is positive, thenÂ Â a/c < b/c

In other words, multiplying or dividing the same POSITIVE number to both sides of an inequality does not change the inequality.

Example 3:Â Solve, write your answer in interval notation and graph the solution set

*Inv. of mult. by 3 is div. by 3 Â *Open interval indicating all values less than -3

*Visual showing all numbers less than -3 on the number line

Interval notation:

(- , -3)

Graph:

A-8

3 : Inequalities relations and interval notations.

Multiplication/Division Properties for Inequalities

when multiplying/dividing by a negative value

If a < bÂ ANDÂ c is negative, thenÂ Â ac > bc

If a < bÂ ANDÂ c is negative, thenÂ Â a/c > b/c

In other words, multiplying or dividing the same NEGATIVE number to both sides of an inequality reverses the sign of the inequality.

Example 4:Â Solve, write your answer in interval notation and graph the solution set

*Inv. of div. by -4 is mult. by -4, so reverse inequality sign *Open interval indicating all values less than -20

*Visual showing all numbers less than -20 on the number line

Interval notation:

(- , -20)

Graph:

A-8

3 : Inequalities relations and interval notations.

Example 5:Â Solve, write your answer in interval notation and graph the solution set

*Inv. of mult. by -2 is div. by -2, so reverse inequality sign *Closed interval indicating all values greater than or = -5/2

*Visual showing all numbers greater than or = -5/2 on the number line

[-5/2, Â )

Interval notation:

Graph:

A-8

3 : Inequalities relations and interval notations.

Strategy for Solving a Linear Inequality

Step 1:Â Simplify each side if needed.Â This would involve things like removing ( ), removing fractions, adding like terms, etc.

Step 2:Â Use Add./Sub. Properties to move the variable term on one side and all other terms to the other side.

Step 3: Use Mult./Div. Properties to remove any values that are in front of the variable.

A-8

3 : Inequalities relations and interval notations.

Example 6:Â Solve, write your answer in interval notation and graph the solution set

*Inv. of add. 5 is sub. 5 Â *Inv. of mult. by -2 is div. both sides by -2, so reverse inequality sign

*Open interval indicating all values greater than -3

*Visual showing all numbers greater than -3 on the number line

(-3, Â )

Interval notation:

Graph:

A-8

3 : Inequalities relations and interval notations.

Example 7:Â Solve, write your answer in interval notation and graph the solution set

*Distributive property *Get x terms on one side, constants on the other side *Inv. of sub. 3 is add. by 3

*Open interval indicating all values less than -1/2

*Visual showing all numbers less than -1/2 on the number line.Â

(- , 8 )

Interval notation:

Graph:

A-8

3 : Inequalities relations and interval notations.

Example 8:Â Solve, write your answer in interval notation and graph the solution set

*Mult. both sides by LCD of 6 Â *Get x terms on one side, constants on the other side

*Inv. of add. 3 is sub. by 3

*Inv. of mult. by 10 is div. by 10Â

*Closed interval indicating all values greater than or equal to -3/2

*Visual showing all numbers greater than or equal to -3/2 on the number line.

Interval notation:

(- 3/2 , )

Graph:

A-8

3 : Inequalities relations and interval notations.

Solving a Compound Inequality

A compound linear inequality is one that has two inequalities in one problem.Â

For example, 5 < x + 3 < 10 or -1 < 3x < 5.Â

Example 9:Â Solve, write your answer in interval notation and graph the solution set

*Inv. of add. 2 is sub. by 2 *Apply steps to all three partsÂ Â *All values between -6 and 8, with a closed interval at -6 (including -6)

*Visual showing all numbers between -6 and 8, including -6 on the number line.Â

.Â

Interval notation:

Graph:

A-8

3 : Inequalities relations and interval notations.

Solving an Absolute Value Inequality

Step 1:Â Isolate the absolute value expression.If there is a constant that is on the same side of the inequality that the absolute value expression is but is not inside the absolute value, use inverse operations to isolate the absolute value.Â Step 2:Â Use the definition of absolute value to set up the inequality without absolute values.

A-8

3 : Inequalities relations and interval notations.

Solving an Absolute Value Inequality

Step 3:Â Solve the linear inequalities set up in step 2.

You will solve these linear inequalities just like the ones shown above.

A-8

3 : Inequalities relations and interval notations.

Example 10:Â Solve, write your answer in interval notation and graph the solution set

*Inv. of sub. 4 is add. by 4 *Apply steps to all three parts Â *All values between -3 and 11

*Visual showing all numbers between -3 and 11

Interval notation:

Graph:

A-8

3 : Inequalities relations and interval notations.

Example 11:Â Solve, write your answer in interval notation and graph the solution set

Step 1:Â Isolate the absolute value expression.Â The absolute value expression is already isolated.Step 2:Â Use the definition of absolute value to set up the inequality without absolute values.AND

Step 3:Â Solve the linear inequalitiesset up in step 2.

Â Be careful, since the absolute value (the left side) is always positive, and positive values are always greater than negative values, the answer is no solution.Â There is no value that we can put in for x that would make this inequality true.

A-8

3 : Inequalities relations and interval notations.

Example 12:Â Solve, write your answer in interval notation and graph the solution set

First inequality, where it is less than or = to -4

*Inv. of div. by 2 is mult. by 2

*Inv. of mult. by -2 is div. by -2, so reverse inequality sign

A-8

3 : Inequalities relations and interval notations.

OR

*Second inequality, where it is greater than or = to 4

*Inv. of div. by 2 is mult. by 2

*Inv. of mult. by -2 is div. by -2, so reverse inequality signsÂ

Interval notation:

Graph:

A-8

3 : Inequalities relations and interval notations.

Example 13:Â Solve, write your answer in interval notation and graph the solution set

Step 1:Â Isolate the absolute value expression.Â The absolute value expression is already isolated.Step 2:Â Use the definition of absolute value to set up the inequality without absolute values.AND

Step 3:Â Solve the linear inequalitiesset up in step 2.

Â Again, be careful - since the absolute value (the left side) is always positive, and positive values are always greater than negative values, the answer is all real numbers.Â No matter what value you plug in for x, when you take the absolute value the left side will be positive.Â All positive numbers are greater than -2.

A-8

End of the Lecture

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