Unit 2 Linear Equations and Inequalities Test Date: Name: By the end of this unit, you will be able to Use rate of change to solve problems Find the slope of a line Model real-world data with linear equations Write an equation of a line in slope-intercept form, point-slope form, and standard form Graph linear equations Write an equation for lines parallel and perpendicular to a given line Find the inverse of a relation and a linear function Create equations and inequalities in one variable and use them to solve problems Solve linear equations and inequalities in one variable Solve and graph compound inequalities Graph inequalities in one and two variables Write inequalities based on word problems
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Table of Contents Slope of a Line... 5 Linear Functions and Rate of Change... 6 Slope-Intercept Form... 8 Graphing on Your Graphing Calculator... 12 Parallel Lines... 15 Perpendicular Lines... 16 Inverse Linear Functions... 17 Solving Inequalities... 18 Compound Inequalities... 21 Graphing Inequalities in 2 Variables... 23 Graphing Special Inequalities... 24 Writing Inequalities... 25 3
Rate of Change Exploration Think about it When someone says a car is traveling at 60 miles per hour, what does this really mean? Rate of Change: A ratio that describes, on average, how much one quantity changes with respect to a change in another quantity. Another word for the rate of change is. The variable we use for this is. The formula we use to calculate rate of change is: *note: the small letters below the numbers are called subscripts example: we read y1 as y sub 1 for x^, y^ and (x a, y a ) Sprinting What variables are being compared? Savings + Salary What variables are being compared? What is the rate of change? What is the rate of change? Studying and Grades What variables are being compared? Car Value What variables are being compared? What is the rate of change? What is the rate of change? Money in Bank Account What variables are being compared? Salary and Commission What variables are being compared? What is the rate of change? What is the rate of change? 4
Slope of a Line The of a nonvertical line is the ratio of the rise to the run. Rewrite the formula for slope here: ghij gkl = Four types of slope: You can use the slope formula to find the slope of a line when given two points. Example 1 Example 2 ( 2,0) and 1,5 ( 3, 4) and 2, 8 Example 3 ( 3, 1) and 2, 1 Example 4 ( 3,2) and 3, 1 Example 5 Example 6 5
Linear Functions and Rate of Change Linear functions have a which means the slope is between any two pairs of points in the function. Check: Are the following functions linear? x y 2 4 3 8 4 12 5 16 x y 1 1 2 4 3 9 4 16 x y 3 6 8 16 10 20 11 22 Challenge! Find a point when given the slope. Find a value of r so that the line Find a value of r so that the line through (6, 3) and r, 2 has a through 1, 4 and ( 5, r) has a slope of ^. slope of ^. a q Find a value of r so that the line through ( 2, 6) and (r, 4) has a slope of 5. 6
Applications: This graph shows the number of people who visited U.S. theme parks in recent years. What is the rate of change from 2000-2002? What is the rate of change from 2002-2004? What is the rate of change from 2004-2006? In what time interval is the rate of change the greatest? You can also find the average rate of change when a graph is not linear. Just use the same formula for rate of change and use your starting and ending points as x^, y^ and (x a, y a ). Find the average rate of change from 2000 to 2006. Vera uploaded a hilarious cat video to YouTube. The table below shows the number of views the video got (y) as a function of time (x) in hours. x 5 10 20 y 70 180 500 During what time interval did the number of views increase the fastest? What was the average rate of change from 5 to 20 hours after Vera posted the video? 7
Slope-Intercept Form A line in slope-intercept form has the form shown below where m represents the, and b represents the. y = mx + b When given the slope and y-intercept Example 1 m = 2, b = 5 Example 2 m = 1 2, b = 6 Example 3 m = 3, b = 2 Example 4 m = 1, b = 4 3 When given the slope and a point 1. Substitute m, x, and y with their 2. Solve for 3. Rewrite the equation using y = mx + b with only identified as numbers Example 1 Write the equation of a line with slope m = 2 that passes through (4, 7). Example 2 Write the equation of a line with slope m = 1 that passes through ( 2, 5). Example 3 Write the equation of a line with slope m = a q that passes through ( 6, 2). 8
When given two points 1. Find the slope of the line 2. Substitute m, x, and y with their known values. (Choose either point!) 3. Solve for b 4. Rewrite the equation using y = mx + b with only m and b identified as numbers Example 1: Write the equation of a line that passes through 3, 2 and (3, 0). Find the slope: Substitute and solve for b: Answer: Example 2: Write the equation of a line that passes through ( 1, 12) and (4, 8). Find the slope: Substitute and solve for b: Answer: Example 3: Write the equation of a line that passes through (2, 1) and (6, 1). Find the slope: Substitute and solve for b: Answer: 9
Graphing a Linear Equation 1. Get equation into y = mx + b form. 2. Plot b (the y-intercept) on the y-axis. 3. Use slope (rise over run) to plot more points. Example 1 Graph y = 2x + 1 Example 2 Graph x + 2y = 6 Example 3 Graph x + y = 3 Example 4 Graph y = a q x 4 10
Write an Equation Given a Graph 1. Identify the y-intercept (b) and slope (m). 2. Write the equation as y = mx + b. Example 1 Example 2 Example 3 Example 4 Write the Special Equation Given the Graph Example 1 Example 2 11
Graphing on Your Graphing Calculator Helpful Hint before you begin: If the [Clear} key doesn't get you out of any screen you are in, use [2nd] [QUIT]. The [2nd] key activates the commands that are above the keys and are the same color as the [2nd] key. Most of the keys you will be using in the following problems are the ones that are directly below the screen. 1. Press the [X,T,O,n] key. What do you see on the screen? 2. Press the [ Y = ] key. If anything is in there, clear it out. In Y1 =, enter 2x 5. 3. Press the [WINDOW] key. If any of the values are different from the ones listed below, change them, Xmin = 10 Ymin = 10 Xmax = 10 Ymax = 10 Xscl = 1 Yscl = 1 Xres = 1 4. Press the [GRAPH] key. What kind of graph do you see? 5. Press the [ 2nd ] key then the [GRAPH] key. Use the up and down arrows to see more values in this table to help you fill in the table at right, then create an accurate graph of the equation. 6. Press [TRACE]. Use the left and right arrow keys to move the "cursor" along the graph. What information do you see at the bottom of the screen? x y What information do you see at the top of the screen? 12
Point-Slope Form We already learned slope-intercept form. We can use point-slope form when we are given the slope and any one point on the line x^, y^. y y^ = m(x x^) Okay cool, but how do I use it? 1. Find the slope (m) 2. Identify a point (x1, y1) 3. Substitute these values into the formula. 4. Simplify into the form you want. Slope-intercept: y = mx + b Standard: Ax + By = C Examples: Write an equation in slope-intercept form of the line whose slope is m = 2 and passes through the point (6, 3). 1. Find the slope: m = 2. Identify a point: (, ) 3. Substitute these values into the formula: y = (x ) 4. Simplify into slope-intercept form: Find the equation of the line where m = ^ q that passes through ( 3,0). Find the equation of the line that passes through ( 2, 1) and has m = 3. 13
Standard Form: An equation in standard form looks like Ax + By = C, where A, B, and C are all constants. The constant A must be positive, but B and C can be either positive or negative. Write an equation in standard form of the line whose slope is m = a that passes through the point q (1,4). 1. Find the slope: m = 2. Identify a point: (, ) 3. Substitute these values into the formula: y = (x ) 4. Simplify into standard form: Find the equation of the line with m = that qˆ passes through ( 2,3). Find the equation of the line with m = 4 and a y- intercept of 3. Write an equation in standard form of the line through the points 2, 3 and (4, 5). 1. Find the slope: m = 2. Identify a point: (, ) 3. Substitute these values into the formula: y = (x ) 4. Simplify into standard form: The cost of a textbook that Ms. Abels uses in her class was $57.65 in 2005. She ordered more books in 2008 and the price increased to $68.15. Write a linear equation to estimate the cost of a textbook in any year since 2005. Let x represent years since 2005. 14
Parallel Lines Write the equations given in the space below and graph them using your graphing calculator. Sketch (doesn t need to be exact!) the graph of the lines on the coordinate plane. Line A: Line B: Line C: Line D: Line E: You Try! 1. Which equation is parallel to y = 5x - 4? A) y = -5x + 1 B) y = 1/5x C) y = -1/5x + 1 D) y = 5x 3. What is the slope of a line parallel to 6x + 2y = 12? 2. Which equation is parallel to y = ¼ x +1? A) y = 4x + 2 B) y = -4x C) y = - ¼ x D) y = ¼ x + 10 4. Which line is parallel to 5x + 3y = 15? A) m = -3 B) m = -1/3 C) m = 3 D) m = 2 5. Write the equation of a line below that that will be parallel to y = 2 x + 4 3 è 6. Show that your line is parallel by graphing both. A) y = 4 ˆ x B) 6x + 10y = 30 q C) y = ˆ x D) 10x + 6y = 30 q 7. Write the equation of a line below that that will be parallel to 3x - 5y = 15 è 8. Show that your line is parallel by graphing both. 15
Perpendicular Lines Line F: Line G: Line H: Line I: Line J: Opposite Reciprocals Write as a fraction ê Flip ê Change the sign ê 9. Which is perpendicular to y = 7x + 1? A) y = -7x + 1 B) y = 1/7x C) y = -1/7x + 1 D) y = 7x 11. What is the slope of a line that is perpendicular to 6x + 2y = 12? A) m = -3 B) m = -1/3 C) m = 1/3 D) m = 2 13. Write the equation of a line below that that will be perpendicular to: y = 6x 3 è 14. Prove that the line you created is perpendicular by graphing both. 10. Which is perpendicular to y = ¼ x +1? A) y = 4x + 2 B) y = -4x C) y = - ¼ x D) y = ¼ x + 1 12. Which is perpendicular to: 5x + 3y = 15? A) y = 4 ˆ x q B) 6x + 10y = 30 C) y = x qˆ D) 10x + 6y = 30 15. Write the equation of a line that will be perpendicular to: 2x + 4y = 8 è 16. Prove that the line you created is perpendicular by graphing both. 16
Inverse Linear Functions Inverse Relations: If one relation contains the point (a,b), then the inverse relation will contain the point. Easy way to remember: Example 1: Finding the inverse of a relation Find the inverse of each relation. {(-3, 26), (2, 11), (6, -1), (-1, 20)} Find the inverse of each relation. Example 2: Graphing the inverse of a function. Remember: Pick some points on the given graph. Then flip the x and y values, and graph the new points! Example 3: Steps: 1) Replace f(x) with. 2) x and y. 3) Solve for. Find the inverse of the function f x = 3x + 27 Find the inverse of the function Find the inverse of the f x = ˆ x 8 function f(x) = 2200 + 0.05x 17
Solving Inequalities An compares the value of two numbers. When we solve a linear inequality, the solution just one number. The solution is a of values. Greater than: Less than: Greater than or equal to: Less than or equal to: b < 4-2 > p 4 k -5 x 18
When you solve a linear inequality, you will use the inverse operations and algebraic used when solving an equation. x + 5 = 7 x + 5 < 7 p - 4 = -3 p - 4-3 There are There are There is only numbers that work There is only numbers that work number that works in this equation. X number that works in this equation. P in this equation. X can be in this equation. P can be any number has to equal. number less than has to equal. than or.. 1. 4 + x < 9 2. 3 + f > -10 3. -6 k 10 4. 2 + g -5 19
5. 3x < 30 6. 5x + 4 29 7. qˆ x > 9 KEY IDEA There is one major difference between solving equations and solving inequalities. To keep a statement when solving an inequality with or, you must the inequality. 8. 2x < 12 9. 9 x + 17 10. 2 x 4 > 8 20
Compound Inequalities Intersection: AND inequalities. In this case, both inequalities are true. Examples: 1. Graph x > 3 and x 5. 2. Graph x < 2 and x > 7. 3. Solve 7 < z + 2 11 and graph its solution set. 4. Solve 3 < 2x 1 < 5 and graph its solution set. A company is manufacturing an action figure that must be at least 11.2 centimeters and at most 11.4 centimeters tall. Write and graph a compound inequality that describes how tall the action figure can be. Words Height is at least 11.2 cm and height is at most 11.4 cm Inequality You are going shopping with your sister for a homecoming dress. She tells you that she can spend up to $200 on the dress, but doesn t want to spend less than $60. Write and graph a compound inequality that describes how much your sister is willing to spend on the dress. 21
A ski resort has several types of hotel rooms and several types of cabins. The hotel rooms cost at most $89 per night and the cabins cost at least $109 per night. Write and graph a compound inequality that describes the amount that a guest would pay per night at the resort. Words Inequality Cost per night is at most $89 or the cost is at least $109 Union: OR inequalities In this case, the compound inequality is true if at least one inequality is true. Example: Graph n 89. Graph n 109. Now, graph their union. Test your graphs with the Test Point Method! 1. Plot important points (remember to choose open/closed circles wisely). 2. Choose one Test Point (TP) from each section. 3. Test the TP in the original equation. a. True? J b. False? L 4. Shade any regions with J s. Examples 1. Graph x > 4 or x < 3. 2. Graph x < 3 or x < 10. 3. Solve 4k 7 25 or 12 9k 30. Graph the solution set. 4. Solve 2k + 5 < 15 or 5x + 15 > 20. Graph the solution set. 22
Graphing Inequalities in 2 Variables Graphing Inequalities Procedure: 1) Solve for y. 2) Graph the line. Remember: < or > is a line and or is a line. 3) Select any test point that is NOT on the line (you choose). Hint: (, ) is often times the easiest to use when possible! 4) Test the point you choose in the inequality substitution! 5) Shade the appropriate side. 1) 3x y < 2 2) x + 6y 6 TP: TP: 3) 2x + 3y 18 4) 2y 4x > 6 TP: TP: Examples: Writing Inequalities 5) 6) Y-intercept: Slope: Equation: Y-intercept: Slope: Equation: 23
Graphing Special Inequalities Review: Graph the linear equations below. y = 3 x = 2 Now, apply the method you learned for graphing inequalities to graph the following: y 3 x 2 y < 3 x > 2 24
Writing Inequalities Common Mistake: Not using the correct inequality symbol. Review: What does each symbol mean? < > Example: Debbie has at most $60 to spend on clothes. She wants to buy a pair of jeans for $22 and spend some more of her money on t-shirts. Each t-shirt costs $8. Tips! 1. Think about what you DON T know. Anything you don t know becomes a variable. 2. Think about what you DO know. Any numbers you know will become constants or coefficients in the equation. 3. Is there more than one possible solution? 4. If so, what type of inequality symbol do you need? Step 1: What DON T you know? Step 2: What DO you know? Step 3: Is there more than one possible solution? How do you know? Step 4: What type of inequality symbol do you need? Write your inequality! Another Example: The girls soccer team wants to raise at least $2000 to buy new goals. They are selling hot dogs for $1 and sodas for $1.25. How many of each item must they sell to buy the goals? 1. Define your variables. 2. Write an inequality that represents this situation. 25