C ollege A lgebra. 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.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
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
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.
Find the root part first and then take it to the exponential part if possible. 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.
A-6
A-6 : Rational Exponents
Review of Exponential Rules
A-6
A-6 : Rational Exponents
Example 1: Evaluate
*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. .
If your exponent’s numerator is 1, you are basically just looking for the root (the denominator’s exponent).
Our answer is 7 since the square root of 49 is 7.
A-6
A-6 : Rational Exponents
Example 2: Evaluate
*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. Again, I think it is easier to do the root part first if possible.
The numbers will be easier to work with. The cube root of -125 is -5 and (-5) squared is 25.
A-6
A-6 : Rational Exponents
Example 3: Evaluate
*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.
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.
Again, I think it is easier to do the root part first if possible. The numbers will be easier to work with.
The square root of 49/36 is 7/6 and 7/6 cubed is 343/216.
A-6
A-6 : Rational Exponents
Example 4: Simplify
* Multiply like bases, add. Exp
A-6
A-6 : Rational Exponents
Example 5: Simplify
*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: Simplify
* Divide like bases, sub. exp
A-6
A-6 : Rational Exponents
Example 7: Simplify 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:
A-8
2 : Solving linear equations.
Equation
Two expressions set equal to each other
Linear Equation
An equation that can be written in the form 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 Equation 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: Solve 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: Solve 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
A-8
2 : Solving linear equations.
Example 3: Solve 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: Solve 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: Solve 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: Solve for x.
Where did our variable, x, go??? It disappeared on us. Also note how we ended up with a TRUE statement, 14 does indeed equal 14. This does not mean that x = 14.
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: Solve 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: Solve for x.
Example 8: Solve for x.
Where did our variable, x, go??? It disappeared on us. Also note how we ended up with a FALSE statement, -2 is not equal to 5. This does not mean that x = -2 or x = 5.
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??? It disappeared on us. Also note how we ended up with a FALSE statement, -2 is not equal to 5. This does not mean that x = -2 or x = 5.
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
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 < b a is less than b a < b a is less than or equal to b
a > b a is greater than b a > b 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. 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. Note that in the interval notations (found below), you will see the symbol , which means infinity.
Positive infinity () means it goes on and on indefinitely to the right of the number - there is no endpoint on the right. Negative infinity (-) means it goes on and on indefinitely to the left of the number - there is no endpoint to the left.
Since we don’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). You will either use a curved end ( or ) or a boxed end [ or ], 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. You still follow the basic ideas described above. The closed end will have a [ or ] on it’s end and the open end will have a ( or ) on its end.
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 – c
In other words, adding or subtracting the same expression to both sides of an inequality does not change the inequality.
Example 1: 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: Solve, write your answer in interval notation and graph the solution set
*Inv. of add 4 is sub. 4 *Closed interval indicating all values greater than or = -3
*Visual showing all numbers greater than or = to -3 on the number line.
[-3, )
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 < b AND c is positive, then ac < bc
If a < b AND c is positive, then 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: Solve, write your answer in interval notation and graph the solution set
*Inv. of mult. by 3 is div. by 3 *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 < b AND c is negative, then ac > bc
If a < b AND c is negative, then 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: 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: 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, )
Interval notation:
Graph:
A-8
3 : Inequalities relations and interval notations.
Strategy for Solving a Linear Inequality
Step 1: Simplify each side if needed. This would involve things like removing ( ), removing fractions, adding like terms, etc.
Step 2: 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: Solve, write your answer in interval notation and graph the solution set
*Inv. of add. 5 is sub. 5 *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, )
Interval notation:
Graph:
A-8
3 : Inequalities relations and interval notations.
Example 7: 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.
(- , 8 )
Interval notation:
Graph:
A-8
3 : Inequalities relations and interval notations.
Example 8: Solve, write your answer in interval notation and graph the solution set
*Mult. both sides by LCD of 6 *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 10
*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.
For example, 5 < x + 3 < 10 or -1 < 3x < 5.
Example 9: 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 parts *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.
.
Interval notation:
Graph:
A-8
3 : Inequalities relations and interval notations.
Solving an Absolute Value Inequality
Step 1: 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. Step 2: 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: 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: 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 *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: Solve, write your answer in interval notation and graph the solution set
Step 1: Isolate the absolute value expression. The absolute value expression is already isolated.Step 2: Use the definition of absolute value to set up the inequality without absolute values.AND
Step 3: Solve the linear inequalitiesset up in step 2.
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. 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: 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 signs
Interval notation:
Graph:
A-8
3 : Inequalities relations and interval notations.
Example 13: Solve, write your answer in interval notation and graph the solution set
Step 1: Isolate the absolute value expression. The absolute value expression is already isolated.Step 2: Use the definition of absolute value to set up the inequality without absolute values.AND
Step 3: Solve the linear inequalitiesset up in step 2.
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. No matter what value you plug in for x, when you take the absolute value the left side will be positive. All positive numbers are greater than -2.
A-8
End of the Lecture
Let Learning Continue