Liberal Arts Math Semester 1 Exam Review

Rules to combine two numbers: When the signs are the same, add and keep the sign. 3 + 5 = 8 -3 + -5 = -8 -3 -5 = -8 When the signs are different, subtract and take the sign of the larger number. 3 + -5 = -2 -3 + 5 = 2 5 + -3 = 2 -5 + 3 = -2 Solve for x: x + (-4) = 10 Solve for x: x – 8 = -5 Solve for x: x + (-12) = 16 Solve for x: x – (-33) = 57 Solve for x: x – (-9) = 24

ex. add vs. subtract and multiply vs. divide. Get the Unknown Alone: The task is always to isolate the variable -- get the variable ALONE on one side of the equal sign. We must do the opposite of what we see, ex. add vs. subtract and multiply vs. divide.    x + 3 = 8 The variable is x and we need to get it alone.      -3   -3 In the problem, 3 is being added to the variable,    x    = 5 so to get rid of the added 3, we do the opposite, subtract 3. In an equation which has more than one operation, we have to  undo the operations in the correct order.  First, undo addition or subtraction, then undo multiplication or division.   5x - 2 = 13 The question is multiplying x by 5, and then subtracting 2.        +2    +2 First, undo the subtraction by adding 2.      5x   = 15           5x  =  15 Then, undo the multiplication by dividing by 5.       5         5        x   =  3

Solve for x: -3x + 6 = 12 Solve for x: -9x – 13 = -103 Solve for x: 8 – 3x = 20 Solve for x: -18 = 4x – 6

Solving an Equation with the Distribution Property

Solve for x: 4(5x – 1) = 2(10x – 2) Solve for x: 5(x + 4) + 2x = 7x + 2(x + 8) Solve for x: 6x – (3 – x) = 3(x + 6) – 1

LCM of 1 and 1 is 6, because both are evenly 3 6 divided by 6 To solve an equation with fractions the first step requires clearing out the fractions! Each term of the equation must be multiplied by the LCM of the denominator to clear out the fractions. (Remember that terms are separated by + and – signs) LCM of 1 and 1 is 6, because 2 x 3 = 6 2 3 LCM of 1 and 2 is 15, because 3 x 5 = 15 3 5 LCM of 1 and 3 is 14, because 2 x 7 = 14 2 7 LCM of 1 and 1 is 6, because both are evenly 3 6 divided by 6

Solving Absolute Value Equations Solve | x | = 5…x could equal 5 or -5 Solve | x + 2 | = 7 There are two possible solutions: (x + 2) = 7      or   (x + 2) = –7  x + 2 = 7        or     x + 2 = –7  x = 5              or     x =  –9 Then the solution is x = 5, –9.

Isolate and Solve the Absolute Value Solve |2x - 1| + 3 = 6  Step 1: Isolate the absolute value |2x - 1| + 3 = 6 (subtract 3 both sides) |2x - 1| = 3 Step 2: Is the number on the other side of the equation negative? No, it’s a positive number, 3, so continue on to step 3 Step 3: Write two equations without absolute value bars 2x - 1 = 3 2x - 1 = -3 Step 4: Solve both equations 2x = 4 x = 2 2x = -2 x = -1 In Step 2, if the number on the other side of the equal sign away from the variable is negative, then the equation has “no solution”

Literal Equations: Equations w/ many variables Solve the equation for one of the variables, so that the one variable you are solving for stands alone. Solve the equation for the letter c: b = cd Solve the equation for the letter x: y = wx z

Solve the equation for the letter y: 3x + 7y = 5

Number Line Inequalities Solve an Inequality

x < -2 The graph of the solution to the inequality: 5x – 2 > 13 Step 1: Solve inequality (add 2 and divide by 5) x > 3 Step 2: Open or closed circle? (open) Step 3: Shaded to the left or to the right? (right) The graph of the solution to the inequality: -6x – 9 > 3 Step1: Solve inequality (add 9 and divide by -6) ***Alert***When multiplying or dividing an inequality by a negative number, you must flip the inequality. x < -2 Step 3: Shaded to the left or to the right? (left)

Solving Inequalities Practice: Solve for x: 6(2x + 4) > 2(3x + 5) Solve for x: -9x – 4 – 5x > -7x – 16

Compound Inequalities: “AND” vs. “OR” disjunction “AND” conjunction

Solve & describe this inequality: 3x – 7 > -10 and 5x + 2 < 22 Solve & describe this inequality: 3x – 7 < -10 or 5x + 2 > 22 Solve & describe this inequality: -6x - 3 > 3 or 2x - 3 > 7

Coordinate Plane Inequalities Greater than or equal to inequality: Solid line shaded above Less than or equal to inequality: Solid line shaded below To graph a greater than inequality: Dashed line shaded above To graph a less than inequality: Dashed line shaded below

Function Terminology In the ordered pair (x, y), the value of x is a member of the ___________. In the ordered pair (x, y), the value of y is a member of the ___________. A relation where every x-value produces a unique y-value is called a ___________. To determine if the graph of a relation is a function, you could use the ____________ test. A set of ordered pairs is called a ____________.

Find the inverse of the function: y = 3x + 2 Invert the variables, i.e. reverse the x and the y x = 3y + 2 Isolate the “y”, i.e. get the y all alone. Subtract 2 from both sides of the equation. Divide both sides of the equation by 3. x – 2 = 3y x – 2 = y 3 Find the inverse of the function y = x + 4 3 Step 1: x = y + 4 Step 2: 3x = y + 4 Step 3: 3x - 4 = y

y = any number is a horizontal line The 4 Types of Slopes  Positive Slope  Negative Slope y = 3x y = 7x + 1 y = 4x -2 5 y = -2x y = -5x + 2 y = -2 x-4 5  Undefined Slope  Zero Slope x = 3 x = -5 x = 2/3 x = any number is a vertical line y= 5 y= -2 y= -5/7 y = any number is a horizontal line

If you graph y = 3x, it is a line with a positive slope (up 3, to the right 1). If you graph y = -5/7x, it is a line with a negative slope (down 5, to the right 7). If you graph x = 2, it is a vertical line with an undefined slope. (There is no "run", creating a zero denominator (ex. 5/0).) If you graph y = 7, it is a horizontal line with a zero slope. (There is no "rise", creating a zero numerator (0/5).)

Point-Slope Form: y - y1 = m ( x - x1) Slope Intercept Form: y = mx + b Find the slope and y-intercept for the equation : 3y = -9x + 15 To put an equation into slope-intercept form, you must isolate the “y” on one side of the equal sign. First solve for "y =“ by dividing all three terms by 3:     3y = -9x + 15 3 3 3 Answer:  y = -3x + 5 the slope (m) is -3                 the y-intercept (b) is 5  Find the equation of the line whose slope is 4 and crosses the y-axis at (0,2). In this problem m = 4 and b = 2. Use the form:   y = mx + b Substitute:            y = 4x + 2     Point-Slope Form: y - y1 = m ( x - x1) Given that the slope of a line is -3 and the line passes through the point (-2,4), write the equation of the line. The slope:  m = -3 The point (x1 ,y1) = (-2,4) Use the form:   y - y1 = m ( x - x1)                       y - 4 = -3 (x - (-2))                       y - 4 = -3 ( x + 2)                         y - 4 = -3x - 6 Answer:  y = -3x - 2

What steps would you take to graph this line, which is shown in slope-intercept form? y = 2x + 3 Step 1: Slope equals 2 which means up 2, to the right 1 Step 2: This line intersects the y-axis at +3, or the ordered pair/point (0,3).

Parallel lines have equal slopes y = 3x + 6 y = 3x - 17 These lines are parallel, because they both have a slope of 3/1 or 3. y = 5x + 8 y = 4x + 8 These lines are NOT parallel, because they have different slopes of 5/1 or 5 and 4/1 or 4. Perpendicular lines have slopes that are negative reciprocals…that means you have to flip the numbers and flip the signs! These lines are perpendicular, because their slopes are negative reciprocals. The first line has a slope of + 3/5, and the second line has a slope of – 5/3. y = 3/5x + 6 y = -5/3x + 2

Find the x and y intercepts of the equation 3x + 4y = 12. The x-intercept of a line is the point at which the line crosses the x axis. ( i.e. where the y value equals 0 ) x-intercept = ( x, 0 ) The y-intercept of a line is the point at which the line crosses the y axis. ( i.e. where the x value equals 0 ) y-intercept = ( 0, y ) Find the x and y intercepts of the equation 3x + 4y = 12. To find the x intercept, plug in zero for the y and solve for x. 3x + 4(0) = 12 3x + 0 = 12 (4,0) 3x = 12 x = 4 To find the y intercept, plug in zero for the x and solve for y. 3(0) + 4y = 12 0 + 4y = 12 4y = 12 (0, 3) y = 3

System of Two Linear Equations Intersecting Lines Parallel Lines Same Line 3x + 2y = 12 3x + 2y = 6 6x - 3y = 12 2x - y = 4 2x + 2y = 6 4x – 6y = 12 One solution: (3, 0) No solution: Parallel lines never touch Infinite solutions: Two lines that are the exact same line and touch each other at every point 2(3) + 2(0) = 6 4(3) – 6(0) = 12 6 + 0 = 6 12 – 0 = 12

What is the solution to the following system? 3x - y = 5 3x - y = 7 3x + 2y = 4 6x + 4y = 8 x + y = 5 3x – y = 7

System of Equations: Word Problem

Types and Degrees of Polynomials What type polynomial is 3x^2-5x+2 What type of polynomial is 6x^4y What type of polynomial is 7x+8y What degree is this? 4x^2 + 5x^3y^2z 9x^2y^3 - 13x^4y^3

Greatest Common Factor (GCF)

Polynomial Addition and Subtraction Simplify: (3x2+3x+5)-(2x2-x-2)= Simplify: (-3x-4)+(2x+6)=

Multiplying Monomials (2x2y)(3x4y2) Original problem (2x2y)(3x4y2) = 6 Multiply the coefficient first. (2 x 3 = 6) Multiply the variables with a base of x. **If the bases are the same, add the exponent (2+4=6) (2x2y)(3x4y2) = 6x6 (2x2y)(3x4y2) = 6x6y3 Multiply the variables with a base of y. **If the bases are the same, add the exponents. (1+2=3) 6x6y3 Final answer

Multiplying Monomials Practice Simplify: (3ab2)(-2a2)(4b) = Simplify: (5ab3)(-a)(-3a2) = Simplify: (-3y)2(5x2) = Simplify: (-4y)3(2x2) = ***Do this exponent first!!!

FOIL Method: First Outer Inner Last

Simplify: (2x+3y)(3x-2y)= Simplify: (3m+2)(m+5)= Simplify: (4k-3)(k-6)= Simplify: (2x+3y)(3x-2y)= ****These are foil problems too!!! Simplify: (4x-y)2= Simplify: (3x+2y)2=

Polynomial Distribution Polynomial Division Polynomial Distribution