2.1 Combining Like Terms

1. 2.1 Combining Like Terms Let’s review some terminology: Variables-Letters that represent numbers Expressions-collection of numbers, variables, grouping symbols, and operations symbols Examples of Expressions: 5 4x-3 2(x+5) Terms-are separated by addition or subtraction signs 3x – 6 terms are 3x and -6 Numerical Coefficient-is the number in front of the variable in 4x the coefficient is 4 in x the coefficient is 1

2. 2.1 Combining Like Terms Constant Term -a number term without a variable in 4x + 5 the constant is 5 Like Terms – have same variables and the same exponents on those variables 3x and x 5xy and 2xy 5 and -3

3. Remember the distributive property? a(b+c) = ab + ac We will use this to eliminate parenthesis by distributing and combining like terms 5(x + 4) -2(3 - x) + 7 5 – (3x – 4) 5x + 20 -6 + 2x + 7 5 – 1(3x - 4) 2x + 1 5 – 3x + 4 9 – 3x

4. 2.1 Combining Like Terms To simplify an expression, use the distributive property to eliminate parenthesis and then combine like terms. Remember Factors are multiplied; Terms are added You cannot solve expressions. You can only simplify them.

5. 2.2 The Addition Property of Equality Expressions set equal to each other are called equations. Solution-when a solution is substituted for the variable in an equation it makes a true statement. To check a possible solution, substitute it into the equation and see if it is true or not. Addition Property of Equality If a = b then a + c = b + c If you have two things that are equal, you can add the same thing to both sides and the two things will remain equal. We use this to solve equations-to isolate the variable:

6. X + 8 = 15 + -8 + -8 x = 7 what could you add to both sides of the equation to isolate the variable? In other words, what would create a zero where the 8 is? Add a -8 to both sides This eliminates the constant (8) and isolates the variable Your solution is 7 2.2 The Addition Property of Equality

7. 2.3 The Multiplication Property of Equality If a = b then ac = bc If you have two things that are equal, you can multiply both sides by the same thing and the two things will remain equal. We use this to solve equations also: Remember that a reciprocal is the same as the multiplicative inverse. 3 1/3 ½ 2 You can isolate the variable by multiplying by the reciprocal of the coefficient OR by dividing by the coefficient itself.

8. dividing by the coefficient itself 3x = 24 3 3 X = 8 multiplying by the reciprocal of the coefficient 3x = 24 (3x) = (24) x = 8 2.3 The Multiplication Property of Equality

9. 2.4 Solve Equations - variable on one side The following steps will work for all equations. Isolate the variable using the following steps: • Use distributive property to remove parenthesis 2(x + 2) = 2x + 4 2. Clear fractions by multiplying both sides by the LCD (optional) 3. Combine Like Terms on each side of equation 4. Use addition property of equality to get all the variable terms on one side of the equation and all the constant terms on the other side 5. Use multiplication property of equality to get rid of the coefficient on the variable 6. Check your solution (optional)

10. 2.4 Solve Equations - variable on one side Remember these terms: Evaluate-find a numerical value 3(4) – 9 (1-3) 5 Simplify-perform the operations and combine like terms 2(x – 5) + 7 (x+ 2) Solve-find the value that makes the equation true 2x + 7 = 15 -7 -7 2x = 8 2 2 x = 4 Check-substitute the value into the original equation to see if it is true: 2x + 7 = 15 2(4) + 7 = 15 8 + 7 = 15 15 = 15

11. 2(x + 3) = 2x + 6 2x + 6 = 2x + 6 -2x -2x 6 = 6 -6 -6 0=0 Distribute Both sides match Everything drops out Solution: Any real number 2.5 Identities and Contradictions Identities-true for an infinite number of solutions

12. 2(x + 5) = 2x + 6 2x + 10 = 2x + 6 -2x -2x 10 = 6 -6 -6 4=0 Distribute Variable terms match Variable drops out What you are left with is NOT true There is NO solution 2.5 Identities and Contradictions Contradictions- have no solution

13. 2.6 Formulas Formulas are commonly used equations that express a specific mathematic relationship Examples: A = l x w or i = prt Evaluate-means to substitute in the given numeric values for the appropriate variables; perform the indicated operations. Find a “value.”

14. 2.6 Formulas-simple interest i = prt (interest = principal x rate x time) Interest=money earned by an investment OR money charged for borrowing money Principal=the amount of money invested or borrowed Rate=the percentage rate expressed as a decimal ( 5% = 0.05 or 7.5% = 0.075 ) Time=the time the money was invested or borrowed IN YEARS (3 months = ¼ year)

15. 2.6 Formulas-simple interest If I borrow $10,000 for 3 years at a 5% APR, how much will I pay in interest? i = p r t i = 10,000 x 0.05 x 3 i = 1500 $1500 will be paid in simple interest

16. 2.6 Formulas-simple interest If I invest $7,000 for 10 years at an unknown interest rate, and earn $4200 in interest, what was the interest rate that I earned? i = p r t 4200 = 7,000 x r x 10 4200 = 70,000 x r 70,000 70,000 0.06 = r The rate of return was 6 %.

17. 2.6 Formulas-geometric applications You will find a variety of geometric formulas in your book. One chart gives the 2-d formulas for perimeter and area. Another gives the formulas for a circle (circumference and area), and a third gives the 3-d formulas. You do not need to memorize these formulas, but you do need to know how to use them. Remember: Perimeter is the distance around a 2-d object (like fencing a yard);measured in regular units Area is the amount of space inside a 2-d object (like painting a wall); measured in square units

18. 2.6 Formulas-geometric applications The Monesteros have an in-ground pool that they would like to fence in. The area needing fencing is 30 feet by 50 feet – rectangular. Find the amount of fence required to enclose all four sides of the pool area. P = 2 L + 2 W P = 2 (50) + 2 (30) P = 100 + 60 P = 160 The perimeter or amount of fence needed is 160 feet.

19. 2.6 Formulas-geometric applications The Monesteros have an in-ground pool that they would like to fence in. The area needing fencing is 30 feet by 50 feet – rectangular. Find the area that is now enclosed by the new fence. A = L x W A = 50 x 30 A = 1500 The area inside the new fence is 1500 sq feet.

20. 2.6 Formulas-geometric applications The Monesteros recently experienced an alien space craft landing in their yard. The diameter of the ring left in the grass is six meters. Find the circumference and area of the ring. C = 2 πr A = π C = 2 x 3.14 x 3 A = 3.14 x 3 x 3 C = 18.85 meters A = 28.3 sq m

21. 2.6 Formulas-solve for a specified variable Solve for W Solve for t P = 2L + 2W i = p r t -2L -2L i = p r t P-2L = 2W pr pr 2 2 P-2L= W i= t 2 pr P – L = W 2

22. Solve for y 2x + 3y = 12 -2x -2x 3y = 12 – 2x 3 3 y = 4 – 2x 3 y = 12 – 2x 3 Given a value for x, find y in the previous example. If x = 0, then y = 12 – 2(0) = 12=4 3 3 2.6 Formulas-solve for a specified variable

23. To solve inequalities, we use the same properties that we used to solve equations EXCEPT when you are eliminating the coefficient, if you multiply or divide by a negative number, you must FLIP the inequality sign. - 3x – 5 10 +5 +5 -3x15 -3 -3 x -5 2.8 Inequalities in One Variable

24. 2.8 Inequalities in One Variable To graph the solution of an inequality, we use a number line and we shade according to the inequality sign. For example: strictly less than and strictly greater than signs will use an open circle to indicate that the point itself is not included in the graph. Less than or equal to and greater than or equal to signs will use a closed circle to indicate that the point itself is included in the graph. In general, if the inequality is written with the variable on the left, the arrow of the inequality sign will indicate the direction you need to shade. (less than to the left; greater than to the right)

25. 2.8 Inequalities in One Variable closed circle < > open circle Remember our solution from the example we did? X ≥ -5 Let’s graph that on a number line: