Linear Equations in One Variable (1.1) Adding in an equation (Objective #1) An equation is a statement involving an equal sign or an expression that is equal to another expression. Add a constant value to both sides of an equation if the equation is in the following form: variable constant difference constant constant EXAMPLE #1: variable difference constant Subtracting in an equation (Objective #2) An equation is a statement involving an equal sign or an expression that is equal to another expression. Subtract a constant value to both sides of an equation if the equation is in the following form: variable constant sum or total constant constant EXAMPLE #2: variable difference constant Multiplying in an equation (Objective #3) An equation is a statement involving an equal sign or an expression that is equal to another expression. Multiply a constant value to both sides of an equation if the equation is in the following form: variable quotient constant variable constant quotient constant EXAMPLE #3: constant variable quotient constant Dividing in an equation (Objective #4) An equation is a statement involving an equal sign or an expression that is equal to another expression. Divide a constant value to both sides of an equation if the equation is in the following form: variable constant product variable constant product constant constant EXAMPLE #4: variable product constant Combining adding, subtracting, multiplying, or dividing to solve a linear equation (Objective #5) A linear equation is a statement that can be written in this form: or (Note: can NOT equal 0 and are constants and is a variable) To write any equation in the above form use the following steps: 1. First remove the brackets [ ] followed by parenthesis ( ) by multiplying. 2. Separate the x terms and the constant terms by switching the signs or adding or subtracting them to both sides of the equation then simplify. 3. Divide both sides by. (Note: If then there is NO SOLUTION. If then there is MANY SOLUTIONS) EXAMPLE #5:
1.1 Linear Equations in One Variable (Objective #1) Solve by adding. (Objective #1) Solve by adding. (Objective #2) Solve by subtracting. (Objective #2) Solve by subtracting. (Objective #3) Solve by dividing. (Objective #3) Solve by dividing. (Objective #3) Solve by dividing. (Objective #4) Solve by multiplying.
1.2 Linear Eqautions in One Variable
Linear Inequalities in One Variable (1.5) Graph linear inequalities using a number line (Objective #1) A linear inequality is a statement involving the following symbols: I always eat the bigger number!!! To graph a linear inequality use a number line and apply the following rules: Inequality x > a x < a x a x a EXAMPLE #1: x 3 Write the solution set of linear inequality using set builder notation (Objective #2) A solution set is a collection of all the solutions for the inequality and written out as follows: EXAMPLE #2: x > 8 Write the solution set of linear inequality using interval notation (Objective #3) A solution set can also be written using interval notation, an interval is a segment of values written as follows where brackets [ means the end value is included and ( means the end value is NOT included: (Note: negative infinity and positive infinity is NOT ever included) EXAMPLE #3: x 3 EXAMPLE #4: x > 2 Solve a linear inequality (Objective #4) Solve a linear inequality the same way as a linear equation. The goal is to write the linear equation in the following form: a x > b + c or a x < b + c or a x b + c or a x b + c (Note: a can NOT equal 0 and a, b, c are constants and x is a variable) To write any inequality in the above form use the following steps: 1. First remove the brackets [ ] followed by parenthesis ( ) by multiplying or clear all fractions or decimals, then remove the brackets followed by the parenthesis. 2. Separate the x terms and the constant terms by switching the signs or adding or subtracting them to both sides of the equation then simplify. 3. Divide both sides by a and flip the inequality if a is negative. EXAMPLE #5:
1.5 Linear Inequalities in One Variable (Objective #1) Graph the inequality. (Objective #1) Graph the inequality. (Objectives #2 & #3) Write the solution set using set builder notation and interval notation. (Objectives #2 & #3) Write the solution set using set builder notation and interval notation. (Objective #3) Solve. (Objective #3) Solve. (Objective #4) Solve. (Objective #4) Solve.
Absolute Value Equations and Inequalities (1.7) Solving Absolute Value Equations (Objective #1) An absolute value represents the distance from the number zero and symbolized and graphed as follows: Use the following steps to solve an absolute value equation: STEPS: 1. Isolate the absolute value in order to determine the distance from zero which is the constant value. 2. Since the number line has a positive side and a negative side there will always be two values on the number line that will produce the same distance from zero, therefore write one equation where the expression inside the absolute value bars equal a positive constant value and another equation where the same expression inside the absolute value bars equal a negative constant value. (Note: If the distance from zero is 0, then there is only 1 solution. If the distance from zero is a negative constant, then there is NO SOLUTION) 3. Solve for the variable in each equation separately. EXAMPLE #2: 5 Solving Absolute Value Inequalities (Objective #2) Use the graph of absolute value inequalities below to solve: or a distance from zero a can be any real number on the number line but the distance from zero is a positive value. EXAMPLE #1: 2 and 2 or k x k and k x k k > x > k k < x < k Use the following steps to solve an absolute value inequality: STEPS: 1. Isolate the absolute value in order to determine the distance from zero which is the constant value. 2. Since the number line has a positive side and a negative side there will always be infinitely many values on the number line that will produce a distance greater than or less than the given distance, therefore use the chart above to determine which compound inequality applies and simply replace with the expression inside the absolute value bars and replace with the constant value. 3. Solve for the variable by leaving the variable in between the two constant values. (Note: If the distance is 0 or negative and or >, then the solution is ALL REAL NUMBERS or MANY SOLUTIONS except 0 if there is >. If the distance is 0 or negative and or <, then there is NO SOLUTION, unless the distance is 0 and then there is only ONE solution) EXAMPLE #3:
1.7 Absolute Value Equations and Inequalities (Objective #1) Solve absolute value equations. (Objective #1) Solve absolute value equations. (Objective #1) Solve absolute value equations. (Objective #1) Solve absolute value equations. (Objective #2) Solve absolute value inequalities. (Objective #2) Solve absolute value inequalities. (Objective #2) Solve absolute value inequalities. > (Objective #2) Solve absolute value inequalities. <
Rectangular Coordinate System (2.1) Plot points on a Cartesian coordinate system or rectangular coordinate system and identify the quadrant in which a point is plotted (Objective #1) A Cartesian coordinate system is made up of two perpendicular number lines that intersect at 0 and divides the plane into four quadrants as follows: ( ) ( ) EXAMPLE #1: Point Plot: (x y) Quadrant #: (3 2) ( 3 4) ( 2 1) (0 4) ( ) ( ) To plot a point find each number on the individual number lines and draw a dot or colored circle where the two numbers meet or intersect and then identify the quadrant the coordinate or ordered pair using romans numerals: I, II, III, or IV. If the dot is plotted on a line then we say NOT on a quadrant OR NONE Graph an oblique line in two variables by plotting points (Objective #2) Oblique lines are slanted lines where both and variables appear in the linear equation as follows: where. To graph an oblique line create a T-Chart and replace any number for and solve for, then plot the points on a Cartesian coordinate system and connect the points using a solid line. (Note: Clear all fractions and decimals) EXAMPLE #2: 2 x y 1 Graph a horizontal line given (Objective #3) Horizontal lines are lines that run east and west or left and right. NO variable will appear in the equation such as: where. To graph a horizontal line, solve the equation for, find the value on the y line and draw a solid horizontal line through the value. (see below for example) Graph a vertical line (Objective #4) Vertical lines are lines that run north and south or up and down. NO variable will appear in the equation such as: where. To graph a vertical line, solve the equation for find the value on the x line and draw a solid vertical line through the value. (see below for example) Find the x and y intercepts given a linear equation (Objective #5) The x-intercept is the 1 st value in a pair where the 2 nd value or is 0. Replace 0 for y and solve for x. The y-intercept is the 2 nd value in a pair where the 1 st value or is 0. Replace 0 for x and solve for y. (see below for example)
Linear Equations In Two Variables (2.1) (Objective #1) Plot and label quadrant #. ( ) ( 0) ( 0) ( ) ( 5) Graph. (Objective #2) x 2 y 0 3 (Objective #2) (Objective #3) Graph. Graph. x y (Objective #4) (Objective #5) Graph. Find the x and y intercepts. x y
The Slope of a Line (2.2) Find the slope of a line given two points on the line (Objective #1) A slope of a line is the ratio between the change in a vertical distance (rise) to the change in a horizontal distance (run). To find the slope of a line given two points on a line use the following formula: (Note: ) EXAMPLE #1: (2 4) (5 1) EXAMPLE #2: 1 2 1 (4 1) Find the slope of a line given an equation of the line (Objective #2) To find the slope of a line given an equation of the line solve the equation for and arrange the terms in the follow form: EXAMPLE #3: x y 6 EXAMPLE #4: 2x 3y 4 Graph a line given its slope and a point on the line (Objective #3) To graph a line given its slope and a point on the line begin by plotting the point on a Cartesian plane. From the point apply the rise and go UP if or go DOWN if then run and go RIGHT if or go LEFT if plot the second point and connect the two points to form a solid line. EXAMPLE #5: Graph. Given slope and point ( ) (refer to figure 1) Use slopes to determine whether two lines are parallel, perpendicular or neither (Objective #4) Apply Objectives #1 and #2. Figure 1
(Objective #1) Find slope given two points. ( )( ) 1.7 The Slope of a Line (Objective #1) Find slope given two slopes. ( ) ( ) (Objective #2) Find the slope given an equation. (Objective #2) Find the slope given an equation. (Objective #3) Graph a line given slope and point. ( ) (Objective #3) Graph a line given slope and point. ( ) (Objective #4) Parallel, perpendicular, or either. (Objective #4) Parallel, perpendicular, or either.