Unit 3 - Multi-Step Equations and Inequaliaties
Solve for x: 3/4x + 4 = 28
3/4x + 4 = 28 -4 -4 ------------- 3/4x 0 = 24 multiply by the reciprocal (4/3)(3/4x) = 24(4/3) fraction to get x alone x = (8)(4) simplify (24/3) b/c 24 & 3 x = 32 have common factor of 3
Solve for x: 3x + 6 = 12
3x + 6 = 12 - 6 -6 inverse operation of the +6 -------- 3x 0 = 6 now need to get variable x by itself --- --- 3 3 dividing by 3/3 (inverse operation) x = 2 is the solution
Solve for x: 4(x + 4) = 4x + 4
4(x+4) = 4x + 4 4x + 16 = 4x + 4 -4x -4x ------------- 16 not= 4 NO solution
Solve for x: 4(x-1) = 1/2(x-8)
4(x-1) = 1/2(x-8) 4x-4 = 1/2x -4 -1/2x -1/2x -------------- is this right? 7/2x - 4 = - 4 +4 +4 ---------- 7/2x not= 0 No solution
Solve for n: 5n - 2 = 3n + 6
5n - 2 = 3n +6 -3n -3n ------------- 2n -2 = 0 +6 +2 +2 ----------- 2n 0 = 8 --- --- 2 2 n = 4 check: 5(4) - 2 = 3(4) +6 20-2 = 12+6 18=18
Solve for x: 5x + 2(11-4x) = 82 + x
5x + 2(11-4x) = 82 + x 5x + 22 - 8x = 82 + x 22 -3x = 82 + 1x -1x -1x -------------- 22 - 4x = 82 -22 -22 -------------- 0 -4x = 60 ---- ---- -4 -4 x = -15 check: 5(-15) + 2(11 -4(-15) = 82 -15 -75 + 2(11 + 60) = 82 - 15 -75 + 22 + 120 = 82 - 15 67 = 67
Solve for x: 5x - 2(x-1) = 8
5x - 2(x-1) = 8 5x -2x + 2 = 8 3x + 2 = 8 - 2 -2 --------- 3x 0 = 6 ---- --- 3 3 x = 2 check: 5(2)-2(2-1)=8 5(2)-2(1)=8 10-2=8 8=8
What is meant by "having parentheses"?
We're talking about the Distributive Property when we talk about dealing with the things that have parentheses in PEMDAS. -3(x-1) <-- distribute the NUMBER & ITS SIGN that's outside the parentheses INTO the parentheses
Solve for x: -2(3x+9) + x = 22
-2(3x+9) + x = 22 check: -6x - 18 + 1x = 22 -2(3(-8)+9)+(-8)=22 -5x - 18 = 22 -2(-24+9)-8=22 +18 +18 48-18-8=22 ------------- -5x 0 = 40 ---- ---- -5 -5 x = -40/5 x = -8
Solve for x: -2/5x + 33 = 39
-2/5x + 33 = 39 -33 -33 -------------- -2/5x 0 = 6 (-5/2)(-2/5x) = 6 (-5/2) x = -30/2 x = -15
When you Multiply or Divide each side of an inequality by a Negative number how do you make it a true statement?
The inequality symbol must be Reversed to make it a true statement when you multiply or divide each side of an inequality by a negative number.
Solve for k. 2k + 18 = 9k -3
2k + 18 = 9k -3 -2k -2k --------------- 0 18 = 7k -3 +3 +3 ------------- 21 = 7k 0 --- ---- 7 7 3 = k check: 2(3)+18=9(3)-3 6+18=27-3 24=24
Solve for x: 2x + 3 = 2x + 3
2x + 3 = 2x + 3 -2x -2x ------------- 0 +3 = 0 +3 3 = 3 All solutions (because every number will yield the answer 3=3)
Solve for x: 2x + 3 = 2x + 7
2x + 3 = 2x + 7 -2x -2x -------------- 3 not= 7 NO SOLUTION
Solve for x: 2x - x + 7 = x + 3 + 4
2x - x + 7 = x + 3 + 4 1x + 7 = x + 7 -x -x ---------- 7 = 7 All solutions
Solve for x: 3(x-1) = 2x + 9
3(x-1) = 2x + 9 3x-3 = 2x + 9 -2x -2x ----------- x -3 = 9 +3 +3 -------- x = 12
Solve for x: 3/2 (2x + 6) = 3x + 9
3/2(2x+6) = 3x + 9 (3/2)(2x/1) + (3/2)(6/1) = 3x + 9 3x/1 + 18/2 = 3x + 9 3x + 9 = 3x + 9 -3x -3x ------------- 0 9 = 0 9 9=9 All solutions
Equations that have variables on both sides can have one of three different solutions. What are they?
(1) A specific solution (2) All solutions (3) NO solution
What are the steps to setting up equations from word problems?
(1) Define your variable with "let statements" (2) Write an algebraic equation (3) Solve your equations showing all steps (4) Check to make sure your answer makes sense
What are the steps for solving equations when there are variables on both sides of the equal sign?
(1) Distribute (2) Combine like terms on each side of the equal side (3) Get variables together using inverse operations (4) Add/Subtract (5) Multiply/Divide (6) Solve (7) Check
Solve for x: 10 = 6x + 34
10 = 6x + 34 -34 -34 ------------------- -24 = 6x ---- ---- 6 6 -4 = x
What are the steps for solving equations with like terms & parentheses?:
(1) Distribute first (2) Combine like terms on one side of the equal sign at a time (3) Do addition or subtraction (inverse operations) (4) Do multiplication or division (inverse operations) (5) Solve (6) Check answer
Three types of answers are possible for solving equations with variables on both sides. What are they?
(1) a solution (2) All solutions (3) No solution
When solving 2-step equations, what are the two important rules to remember?
(1) use inverse operations to isolate the variable (move everything AWAY from the variable) (2) SADMEP --> PEMDAS backwards : always do Subtraction or Addition first then do Multiplication and Division
Solve for x: -1/2x - 5 = 12
-1/2x - 5 = 12 +5 +5 ----------- -1/2x 0 = 17 (-1/2x)(-2/1)= 17 (-2/1) x = -34
Solve for x: -x/10 + 5 = -7
-1x/10 + 5 = -7 -5 -5 ---------- -1x/10 0 = -12 (-10/1)(-1x/10) = -12(-10/1) x = 120
Solve for x: -8(2x-1) = 36
-8(2x-1) = 36 -16x + 8 = 36 -8 -8 -------- -16x 0 = 28 ----- ---- -16 -16 x = -28/16 reduce this x = -7/4
Solve for x: -x + 5 = -20
-x + 5 = -20 -5 -5 inverse operation ------------- -x 0 -25 now get x by itself --- ---- divide by -1/1 because -1 -1 can't have negative variable x = 25 is the solution
Solve for x: -x/2 + 4 = 6
-x/2 + 4 = 6 - 4 -4 ------------ -x/2 0 = 2 (-1x/2)(-2/1)= 2(-2/1) x = -4
When a number is divided by itself, what is the answer?
1 --> A number divided by itself is always 1
Solve: 11 + 3x - 7 = 6x + 5
11+3x-7=6x+5 -3x -3x ----------- 11 0 -7 = 3x + 5 -5 -5 ---------------- 6 -7 = 3x -1 = 3x -- --- 3 3 -1/3 = x check: 11 +3(-1/3) - 7 = 6(-1/3) + 5 11 + (-3/3) -7 = (-6/3) + 5 11 + -1 -7 = -2 + 5 3 = 3
Solve for x: 13 = 2y - 3(y+4)
13 = 2y - 3(y+4) 13 = 2y -3y -12 13 = -1y - 12 +12 +12 -------------- 25 = -1y 0 --- --- -1 -1 -25 = y check: 13=2(-25)-3(-25+4) 13=2(-25)-3(-21) 13=-50+63 13=13
Solve for x: x/6 - 8 = -32
1x/6 - 8 = -32 +8 +8 ----------- 1x/6 0 = -24 (6/1)(1x/6)= -24(6/1) x = -144
Solve for y: 6y - 40 = y
6y - 40 = 1y -6y -6y ---- ---- 0 -40 = -5y ---- ----- -5 -5 8 = y check: 6(8)-40=8 48-40=8
Solve for x: 7p + 8p - 12 = 59
7p + 8p -12 = 59 15p -12 = 59 +12 +12 --------- 15p = 71 ---- --- 15 15 p = 71/15 reduce this p = 27/5 check: 7(27/5)+8(27/5)-12=59 189/5+216/5-12=59 81-12=?? (this isn't right)
Solve for x: 8x - 17 = 27 - 3x
8x - 17 = 27 - 3x +3x +3x ------------------ 11x - 17 = 27 +17 +17 --------- 11x 0 = 44 ---- ---- 11 11 x = 4 check: 8(4)-17 = 27-3(4) 32-17 = 27 -12 15 = 15
Solve for m: 9m + 13 = m + 21
9m + 13 = m + 21 -m -m -------------- 8m + 13 = 0 + 21 -13 -13 ----------- 8m = 8 ---- --- 8 8 m = 1 check: 9(1)+13=1+21 9+13=1+21 22=22
What are the inequality symbols?
< less than > greater than <= less than or equal to >= greater than or equal to not=
When graphing inequalities, what does a closed circle mean?
A closed circle means that the number that the circle is above is included in the solution to make the inequality a true statement -- that number can be one of the solutions so the solution is either greater or less than or equal to the number
What is an inequality?
An inequality is a statement formed by placing an inequality symbol between 2 expressions < less than > greater than <= less than or equal to >= greater than or equal to not=
When graphing inequalities, what does an open circle mean?
An open circle means that the number that the circle is above is not included in the solution to make the inequality a true statement -- the solution is either greater or less than that number
What does "All Solutions" mean when solving equations with variables on both sides.
Any number we plug in to the equation for X will work. Example: 4(x+1) = 4x + 4 4x + 4 = 4x + 4 -4x -4x ------------- 0 4 = 0 4 4=4 All Solutions you substitute for x will work
When discussing negative integers, do consecutive numbers move toward the 0 on the number line or away from the zero on the number line?
Consecutive numbers in negative integers count TOWARD the 0 on the number line: -10, -9, -8, -7.... because they are actually getting bigger going that direction, just like positive consecutive numbers get bigger (but move away from 0)
When graphing the solution for an inequality, what direction do you draw the arrow for a solution that is less than?
Draw the arrow pointing to the left for a solution that is less than < even when dealing with negative numbers.
When graphing the solution for an inequality, what direction do you draw the arrow for a solution that is greater than?
Draw the arrow pointing to the right for a solution that is greater than > even when dealing with negative numbers.
Define like terms
Each term has the same variables: 5 & 7 or 2x & 3x 2x and 3xsquared are NOT b/c x and xsquared are not the same variable
What does it mean to use an inverse operation on an equation?
Example: If a number is being subtracted in an equation, you can add the same number to both sides of the equation and the equation remains true. The same is true for the other operations. Inverse operations for each operation are: addition --> subtraction subtraction --> addition multiplication --> division division --> multiplication
What does it mean to use SADMEP when solving equations?
First do the addition and subtraction -- referring to numbers that are not tied to a variable. Then do the multiplication and division that are tied to numbers that are attached to variables by multiplication or division.
Set up an equation and solve: The sum of three consecutive integers is 39. Find the integers.
Let x = what the problem is asking us to do or what we are looking for: find 3 integers let x = 1st integer let x+1 = 2nd integer let x=2 = 3rd integer Use a variable that represents everything that we're looking for so we can solve it. If we used 3 separate variables - 1 for each integer - we'd need a lot more info to solve it. x+x+1+x+2 = 39 3x +3 = 39 -3 -3 ------- 3x = 36 --- --- 3 3 x = 12 x+1 = 13 x+2 = 14 check: 12+13+14 = 39
Set up an equation and solve: The sum of three consecutive odd integers is 45. Find the integers.
We are looking for 3 consecutive odd integers so we need 3 let statements. let x = 1st integer let x+2 = 2nd integer let x+4 = 3rd integer x+x+2+x+4 = 45 3x + 6 = 45 -6 -6 --------- 3x = 39 --- --- 3 3 x = 13 x+2 = 15 x+4 = 17 check: 13+15+17=45
Set up an equation and solve: Find four consecutive integers whose sum is -42.
We are trying to find 4 consecutive integers. let x = 1st integer let x+ 1 = 2nd integer let x +2 = 3rd integer let x + 3 = 4th integer x+x+1+x+2+x+3 = -42 4x +6 = -42 -6 -6 -------- 4x = -48 --- --- 4 4 x = -12 x+1 = -11 x+2 = -10 x+3 = -9 check: -12 + -11 + -10 + - 9 = -42
Set up an equation and solve: The length of a rectangle is 11cm more than the width. The perimeter is 90cm. Find the length and width of the rectangle.
We are trying to find the length and width. We know that the length is the width plus 11cm. Let x = width Let Length = x + 11 we know that perimeter is length plus length plus width plus width or P=2(l+w) P=90 x + x + x+11 + x+11 = 90 OR 90 = 2(x+(x+11)) 4x + 22 = 90 -22 -22 ---------- 4x 0 = 68 --- --- 4 4 x = 17 width length = 17+11 = 28 check: 2(17 + 28) = 90 34+ 56 = 90
Set up an equation and solve: Three more than six times a number is 21. Find the number.
We want to find a number. Let x=a number 6x +3 = 21 -3 -3 ------- 6x = 18 --- --- 6 6 x = 3 check: 6(3) +3 = 21 18+3=21
When do you use PEMDAS?
When evaluating expressions. (Clear the parentheses, exponents, multiplication, division, addition, subtraction) Address the multiplication and division in the order of which operation comes first when reading the expression from left to right. Then address the addition and subtraction in the order of which operation comes first when reading the expression from left to right.
When solving an inequality, how do you treat the inequality sign?
When solving an inequality, solve it like an equation, at first treating the inequality sign as an equal sign for the purposes of isolating variables, etc.
When do you use SADMEP?
When solving equations. Do the addition & subtraction to the terms that are not tied to variables. Then do the multiplication and division to the terms tied to variables.
Is it true that all numbers that are attached to a variable are tied to that variable with either multiplication or division?
YES, if a number is attached to a variable, it is tied to it be either multiplication or division: 5z means 5 times variable z 5/z means 5 divided by variable z
A negative divided by a negative is what sign?
a positive
When solving inequalities, always write the answer so the variable is on the left. Sometimes that requires moving the variable from the right side to the left side. When you have to move the variable from the right side to the left side, make sure to change the inequality sign to reflect the change and keep the original statement.
example: 7 < h - 3 +3 +3 ---------- 10 < h move the h to the other side h > 10 switched the inequality symbol so statement remains the same
When you Multiply or Divide each side of an inequality by a Negative number, what do you have to do to the inequality symbol to keep it a true statement?
flip the symbol
Solve for m: m-1 = 9m +15
m - 1 = 9m + 15 -m -1m ------------ 0 -1 = 8m + 15 -15 -15 -------------- -16 = 8m ---- ---- 8 8 -2 = m check: -2-1 = 9(-2)+15 -3= -18+15 -3 = -3
Solve for x: x/5 - 8 = 6.2
x/5 - 8 = 6.2 6.2 14.2 + 8 +8 +8.0 x 5 -------------- ------ ----- x/5 0 = 14.2 14.2 121.0 (1x/5)(5/1)= 14.2(5/1) x = 121
