2.1 Simplifying Algebraic Expressions

Transcription

1 .1 Simplifying Algebraic Expressions A term is a number or the product of a number and variables raised to powers. The numerical coefficient of a term is the numerical factor. The numerical coefficient of 3x is 3. Example 1 Identify the numerical coefficient of each term: a) 9x b) 3y c) x d).7x y contain the same variables raised to the same powers. Terms that are not like terms are called unlike terms. Like Terms Unlike Terms 3x, x 5x, 5x Example Indicate whether the terms in each list are like or unlike: a) 6x, 3x b) xy, x y c) 5ab, 1 ba d) 3 x yz, 3 3 x yz

2 Eby, MATH 0310 Spring 017 Page Simplifying the sum or difference of like terms is called. To combine like terms add or subtract the numerical coefficient and leave the variable part the SAME! Example 3 Simplify each expression by combining any like terms: a) 7x x+ 4 b) 9y y 7 c) 3x y+ 5xy x y+ xy Sometimes you will first have to simplify using the before you can combine like terms. To use the distributive property multiply each term inside the parentheses by what is in front of them. Be careful with negatives! Example 4 Simplify each expression. Use the distributive property to remove any parentheses. a) ( 5m+ 6n p) b) 1 (6x 9) 3

3 Eby, MATH 0310 Spring 017 Page 3 c) 14(x+ 6) 4 d) 3(x 5) ( x+ 7) And then there will be times that you need to translate from word to algebraic expressions. In order to do so you might need to know what some words mean: + : sum, added, more than, increased by, plus, total of, together - : subtracted from, less than, difference between, decreased, minus x : product, times, any ⅜(etc.) of or % of, multiplied by : divided, quotient, ratio, over, per Example 5 a) Triple a number, decreased by six. b) Six times the sum of a number and two, increased by three.

4 Eby, MATH 0310 Spring 017 Page 4. The Addition and Multiplication Properties of Equality Last time we talked about simplifying expressions, this time we will solve linear equations. Before we do that let s look at the difference between expressions and equations! Expressions Equations A is any equation that can be written in the form: ax+ b= c where a, b, and c are real numbers and a 0. In order to solve linear equations we will need to know some properties! The states that if a, b, and c are numbers and if a= b, then a+ c= b+ c. This is also true for subtraction! So if a= b, then a c= b c. Example 1 Solve each equation: a) y 6= 18 b) 18= x+ 5 c) 3 a+ = 3 4

5 Eby, MATH 0310 Spring 017 Page 5 The states that if a, b, and c are numbers and if a= b, then a c= b c. This is also true for division! So if a= b, then a c= b c. Example Solve each equation: x a) y= 18 b) = 8 3 c) 1 3 a= Very rarely will we use only one property at a time though! Example 3 a) 8( y+ ) = 4( y 3) b) 8z+ 5+ 6z= 3z+ 10 c) x = x

6 Eby, MATH 0310 Spring 017 Page 6 And again there will be times that you need to translate from word to algebraic expressions! Consecutive Integers: e.g.: 1,, 3, 4.Expressed by: x,,, Consecutive Even Integers: e.g.:, 4, 6.Expressed by: x,,, Consecutive Odd Integers: e.g.: 3, 5, 7.STILL: x,,, Example 4 a) If x is the first two consecutive integers, express the sum of the two integers in terms of x. Simplify if possible. b) If x is the first two consecutive odd integers, express the sum of the two integers in terms of x. Simplify if possible. c) Two numbers have a sum of. If one number is z, express the other number in terms of z. d) On a recent car trip, Raymond drove x miles on day one. On day two, he drove 170 miles more than he did on day one. How many miles, in terms of x, did Raymond drive for both days combined?

7 Eby, MATH 0310 Spring 017 Page 7.3 Solving Linear Equations So in the last section we solve linear equations, and it is still mostly the same. Example 1 Solve the following linear equations: a) 6 a (5a 1) = 4 b) 3(a 3) = 5( a+ 4) But now we want to make our lives better by getting rid of the fractions! Steps for Killing Fractions 1. Find the LCD of fractions.. both sides of the equation, parentheses, by the LCD you found in step to kill the fraction! 4. Solve the resulting equation.

8 Eby, MATH 0310 Spring 017 Page 8 Example Solve the following linear equations: a) 1 x 3 x= 5 b) 1 ( x 9) = 5 ( x+ 6) c) 6x+ 5 5x + 1 = 4 4 All of the equations we have solved so far have had one single answer. These types of equations are called. There are also equations that will never be true, they have no solution, and they are called. And sometimes equations will be exactly the same on both sides (they are always true), the solution is, and these types of equations are called. Type Conditional Solution All Real Numbers

9 Eby, MATH 0310 Spring 017 Page 9 The nice thing about math is that it is very easy to tell when an equation is true and when it is not true. Some examples are: True Statements Not True Statements 3=3 3=5 Example 3 Solve each equation. Indicate its type and its solution: a) 6( z+ 7) = 6z+ 4 b) x x 3=

10 Eby, MATH 0310 Spring 017 Page 10.4 Introduction to Problem Solving More Words That Mean Addition ( + ) Subtraction ( - ) Multiplication ( ) Division ( ) Equal Sign (=) Sum Difference of Product Quotient Equals Plus Minus Times Divide Gives Added to subtracted from Multiply Into Is/was/ should be More than Less than Twice Ratio Yields Increased by Decreased by Of Divided by Amounts to Total Represents Is the same as

11 Eby, MATH 0310 Spring 017 Page 11 Example 1 a) Eight is added to a number and the sum is doubled, the result is 11 less than the number. Find the number. b) Three times the difference of a number and is equal to 8 subtracted from twice a number. Find the integers. c) A college graduating class is made up of 450 students. There are 06 more girls than boys. How many boys are in the class?

12 Eby, MATH 0310 Spring 017 Page 1 d) A triangle has three angles, A, B, and C. Angle C is 18 greater than angle B. Angle A is 4 times angle B. What is the measure of each angle? (Hint: The sum of the angles of a triangle is 180 ). e) The room numbers of two adjacent hotel rooms are two consecutive odd numbers. If their sum is 1380, find the hotel room numbers.

13 Eby, MATH 0310 Spring 017 Page 13.5 Formulas and Problem Solving A is an equation that states a known relationship among multiple quantities (has more than one variable in it). Formulas you may need Perimeter: ALWAYS the sum of all of the sides!! Circumference (circle) = π r or π d Area: Square = s Rectangle = l w Triangle = 1 bh or bh Parallelogram = b h Trapezoid = 1 ( ) 1 hb + b Circle = π r Volume: Box = l w h Cylinder = π r h Misc: Distance: D= rt Temperature: 5 C= ( F 3) 9 Interest: I = prt

14 Eby, MATH 0310 Spring 017 Page 14 Sometimes you may need to solve a formula for a different variable than what you start with. Example 1 Solve for the indicated variable: a) 1 A= bh for b b) S = 4lw+ wh for l And other times you ll have to solve an entire word problem using a formula. Example a) You have decided to fence an area of your backyard for your dog. The length of the area is 1 meter less than twice the width. If the perimeter of the area is 70 meters, find the length and width of the rectangular area. b) For the holidays, Chris and Alicia drove 476 miles. They left their house at 7 a.m. and arrived at their destination at 4 p.m. They stopped for 1 hour to rest and re-fuel. What was their average rate of speed?

15 Eby, MATH 0310 Spring 017 Page 15.6 Percents and Problem Solving The most commonly known way to solve percent problems is: = where is is the resulting amount, and of is the original base number of the problem. Example 1 a) 544 is what percent of 640? b) What is 4% of 740? c) 177 is 0% of what number? But of course we don t usually see them this way. Most of the time they will be word problems where it is harder to decide the is and the of. Just remember is is the, and of is the of the problem. Sales Tax: Sales tax is some % of the original price. Price including tax is 100+some % of the original price. Example Patrick paid $91.16 for golf shoes. If the sales tax rate is 6%, what was the price of the shoes?

16 Eby, MATH 0310 Spring 017 Page 16 Discounts: Discount is some % of the original price. Price after discount is 100-some% of the original price. Example 3 A jacket is on sale for $ If this represents a 0% discount, what was the original price of the jacket? Percent increase or decrease Percent change = new amount - original amount 100 original amount If your answer is positive then it is a percent, and if your answer is negative then it is a percent. Example 4 a) The number of minutes on a cell phone bill went from 100 minutes in March to1600 minutes in April. Find the percent increase. Round to the nearest whole percent.

17 Eby, MATH 0310 Spring 017 Page 17 b) In 004, a college campus had 8,900 students enrolled. In 005, the same college campus had 7,600 students enrolled. Find the percent decrease. Round to the nearest whole percent.

18 Eby, MATH 0310 Spring 017 Page 18.7 Further Problem Solving You will also encounter many problems that involve distance, money, and interest. Distance Problems Any time you have a problem that involves distance or travel you will want to use the distance formula:. Example 1 How long will it take a car traveling 60 miles per hour to overtake an activity bus traveling 45 miles per hour if the activity bus left hours before the car? Car Activity Bus D r t Money Problems Whenever you have problems dealing with values then you need to remember that you have a number of things and a value for those things. For example if I have three quarters I would say that the value of a quarter is, and the number of quarters I have is, so the total value of my money is.

19 Eby, MATH 0310 Spring 017 Page 19 Example A collection of dimes and quarters and nickels are emptied from a drink machine. There were four times as many dimes as quarters, and there were ten less nickels than there were quarters. If the value of the coins was $19.50, find the number of quarters, the number of dimes, and the number of nickels. Quarters Value each Number Total Value Solution Dimes Nickels Total Example 3 The local church had an ice cream social, and sold tickets for $3 and $. When the social was over, 81 tickets had been sold totaling $15. How many of each type of ticket did the church sell? $ tickets Value each Number Total Value Solution $3 tickets Total

20 Eby, MATH 0310 Spring 017 Page 0 Interest Problems If a problem talks about saving or investing then you have an interest problem and will need to use the formula:. Example 4 Jeff received a year-end bonus of $80,000. He invested some of this money at 8% and the rest at 10%. If his yearly earned income was $7,300, how much did Jeff invest at 10%? 8% Fund Principle Rate Time Interest 10% Fund Total

21 Eby, MATH 0310 Spring 017 Page 1.8 Solving Linear Inequalities Everything we have done so far have been expressions or equations. Now we will solve inequalities. An inequality is a statement that contains of the symbols: <, >, or. Equation x = 3 Inequality 1 7 3y Graphing and Interval Notation When you graph an inequality on a number line or write the solution in interval notation you need to know some symbols. Symbol ( or ) Meaning [ or ] Example 1 Graph each inequality on a number line and write it in interval notation: a) x 5 b) y< 7 c) 3 m d) x> 5

22 Eby, MATH 0310 Spring 017 Page Set Builder Notation Another way to write the solution to an inequality is to use set builder notation. { } Example Write the solution to each inequality in Set Builder Notation: a) x 5 b) y< 7 c) 3 m d) x> 5 Solving Linear Inequalities in One Variable The nice thing is that all of the properties of equality are the same!! The only thing you need to be careful of is or by a number!

23 Eby, MATH 0310 Spring 017 Page 3 Example 3 Use the properties of equality to solve each of the following inequalities. Graph and write the solution in both set builder and interval notation. a) x+ 7 1 b) y 10> 3 c) x 8 d) 3 y> 4 3 e) 8 1 ( x+ ) > ( x+ 3) f) 18( a ) 1a

24 Eby, MATH 0310 Spring 017 Page 4 Compound Inequalities A compound inequality contains two inequality symbols. This means 0 4(5 x) AND 4(5 x) < 8. To solve the compound inequality, perform operations simultaneously to all three parts of the inequality (left, middle and right). Example 4 Solve each compound inequality. Graph and write the solution in both set builder and interval notation. a) 5< t 0 b) 1 x< 8 c) 3 4x 9 7

25 Eby, MATH 0310 Spring 017 Page 5 Word Problems Since we solve inequalities almost exactly the same as we solve equations, we will solve word problems with inequalities almost exactly the same as we solve word problems with equations. Just be careful with your symbols! Example 5 a) Chris paid more than $00 for a suit. b) A candidate for President must be at least 35. c) The capacity of an elevator is at most 1,500 lbs. d) The company must have no fewer than 10 programmers. e) The difference of four times a number and eight is less than two times the number. f) One side of a triangle is six times as long as another side and the third side is 8 inches long. If the perimeter can be no more than 106 inches, find the maximum lengths of the other two sides. END OF EXAM 1 MATERIAL