Reteaching Transforming Linear Functions

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1 Name Date Class Transforming Linear Functions INV 6 You have graphed linear functions on the coordinate plane. Now you will investigate transformations of the parent function for a linear function, f(x) x. Graph f(x) x and f(x) x on a coordinate plane with the parent function f(x) x. Then describe the transformations. Step 1: Graph both functions on a coordinate plane with the parent function. Step : Describe the transformations. The graph of the line f(x) x shifts up units from the graph of the line f(x) x. The y-intercept shifts from (0, 0) to (0, ). The graph of the line f(x) x shifts down units from the graph of the line f(x) x. The y-intercept shifts from (0, 0) to (0, ). f(x) = x + x O y f(x) = x 6 f(x) = x Changing the y-intercept will shift the line up or down. This is called a translation. Complete the steps to describe the transformations. 1. Graph f(x) x 3 and f(x) x on a coordinate plane with the parent function f(x) x. Then describe the transformations. The graph of the line f(x) x 3 shifts up 3 units from the graph of the line f(x) x. The y-intercept shifts from (0, 0) to (0, 3). f(x) = x + 3 x y f(x) = x O 6 - f(x) = x - The graph of the line f(x) x shifts down units from the graph of the line f(x) x. The y-intercept shifts from (0, 0) to (0, ). The transformations are translations.. Graph f(x) x 3 on a coordinate plane with the parent function f(x) x. Then describe the transformation. The graph of the line f(x) x 3 shifts down 3 units from the graph of the line f(x) x. The y-intercept shifts from (0, 0) to (0, 3). The transformation is a translation. -6 x - - O - - y f(x) = x 6 f(x) = x 3 Saxon. All rights reserved. 131 Saxon Algebra 1

2 continued INV 6 Graph f(x) x and f(x) x and on a coordinate plane with the parent function f(x) x. Then describe the transformations. Step 1: Graph both functions on a coordinate plane with the parent function. Step : Describe the transformations. The graph of the line f(x) x is steeper than the graph of the line f(x) x. When the slope changes, it causes a stretch or compression of the graph of the parent function. -6 x - - O - - y f(x) = x f(x) = x 6 f(x) = x The graph of the line f(x) x is steeper than the graph of the line f(x) x and is reflected over the x-axis. The transformation is both a stretch and a reflection. Complete the steps to describe the transformation. y f(x) = 3x 3. Graph f(x) 3x on a coordinate plane with the parent function f(x) x. Then describe the transformation. The graph of the line f(x) 3x is steeper than the graph of the line f(x) x. When the slope changes, O - - f(x) = x 6 it causes a stretch of the graph of the parent function.. Graph f(x) x on a coordinate plane with the parent function f(x) x. Then describe the transformation. The graph of the line f (x) x is a reflection of the line f(x) x over the x-axis. O f(x) =-x y f(x) = x 6 - Saxon. All rights reserved. 13 Saxon Algebra 1

3 Name Date Class Simplify Radical Expressions 61 You have found square roots of perfect squares. Now you will simplify radical expressions using the product of radicals rule. Product of Radicals Rule If a and b are non-negative real numbers, then n n a b n ab and n ab n n a b. Simplify using perfect squares: 6. Step 1: Find factors of 6 that are perfect squares and 16 are perfect squares whose product is 6. Step : Apply the Product of Radicals Rule. 16 Use n ab n n a b. Step 3: Simplify the perfect squares. The square root of is. The square root of 16 is. 8 Multiply. Complete the steps to simplify using perfect squares Simplify using perfect squares Saxon. All rights reserved. 133 Saxon Algebra 1

4 continued 61 The rules of radicals and exponents also apply to variable expressions. Simplify: 100 x y 5. Step 1: Find factors of 100 x y 5 that are perfect squares. 100 x y 5 10 x y y Use Rules of Exponents to write y 5 as y y 1. Step : Apply the Product of Radicals Rule. 10 x y y Step 3: Simplify the perfect squares. 10 x y y 10x y y Use n ab n n a b. Multiply. Complete the steps to simplify each expression x 3 y x y x 3 y 10 x x y 10 x x y 10 x y 5 10 x y y 10 x y y 10 x x y 10 x y y 10x y x x y 10y Simplify each expression x 1. 1 x y 10x 1xy x x 3 y 3 6x x 7xy xy x ,000 y 10x y x 0. 7 x 3 y 5x 3 6xy x 1. Find the length of one side of a square room with an area of 96 square meters. 6 m Saxon. All rights reserved. 13 Saxon Algebra 1

5 Name Date Class Now you will display data in a stem-and-leaf plot. The list below shows the daily low temperatures ( F) for a town in the Northeast. Create a stem-and-leaf plot of the data. 0, 56, 50, 60, 6, 63, 9, 8, 9, 0, 36, 59, 39 Step 1: There are two place values in each temperature in the data set, tens and ones. Organize the data by each tens value. Write each place-value group in ascending order. Include any values that repeat. 30 s: 36, 39 0 s: 0, 0, 8, 9, 9 50 s: 50, 56, s: 60, 6, 63 Displaying Data in Stem-and-Leaf Plots and Histograms 6 Step : Use the tens digit of each group as the stem of a row on a stem-and-leaf plot. Write each ones digit as the leaf for the corresponding tens digit. Step 3: Create a key to show how to read each entry in the plot. Daily Low Temperatures ( F) Stem Leaf 3 6, 9 0, 0, 8, 9, 9 5 0, 6, 9 6 0,, 3 Key: 6 6 F Complete the steps to create a stem-and-leaf plot of the data. 1. The list below shows the daily high temperatures Daily High Temperatures ( F) ( F) for a town in the Northeast. Create a stem-and-leaf plot of the data. 70, 8, 71, 73, 71, 70, 73, 78, 76, 65, 65, 67, 66 Stem Leaf 6 5, 5, 6, 7 7 0, 0, 1, 1, 3, 3, 6, 8 8 Create a stem-and-leaf plot of the data.. The list below shows the test scores from Mr. Clark s History class. 75, 81, 96, 63, 78, 9, 88, 8, 87, 99, 9, 68, 70 Key: F Stem Leaf 6 3, 8 7 0, 5, 8 8 1,, 7, 8 9,, 6, 9 Key: Saxon. All rights reserved. 135 Saxon Algebra 1

6 continued 6 Stem-and-leaf plots are used to find measures of central tendency. Use the stem-and-leaf plot to find the median, mode, range of the data, and the relative frequency of $8.00. Cash Sales for Last 15 Transactions Stem Leaf 6, 7, 7, 8 8 0, 0, 0, 1, 3, 6, 7 9, 5 10 Key: 9 $9.0 Step 1: Find the median. Find the middle value(s). For this data there is one middle value. The 8th value is $8.00. The median of the data is $8.00. Step : Find the mode. The mode is the value or values that occurs the most frequently. There is one value that occurs 3 times: $8.00. The mode of the data is $8.00. Step 3: Find the range. The range is the difference between the greatest and the least value in the set. The greatest data value is $10.0; the least data value is $6.0. The difference is $.00. The range of the data is $.00. Step : Find the relative frequency of $8.00. The relative frequency is the number of times the data value occurs divided by the total number of data values. Number of times $8.00 occurs Total number of data values % Complete the steps to find the median, mode, and range of the data using the stem-and-leaf plot. Find the relative frequency of Stem Leaf 5 The median is The data value 37 occurs most frequently. 0 1, 1, 7 1, 6, 5, 5 3 6, 7, 7, 7 Key: The mode is 37. Find the difference between the greatest and least data value The range is % The relative frequency of 37 is 5%. Saxon. All rights reserved. 136 Saxon Algebra 1

7 Name Date Class Now you will solve systems of linear equations by elimination. Solve the system of equations by elimination and check the solution. 3x y 6 5x y 10 Add the equations to eliminate y. Step 1: Add the equations vertically. 3x y 6 _ 5x y 10 8x 16 Combine like terms. x Divide by 8. Solving Systems of Linear Equations by Elimination 63 Solve the system of equations by elimination. Check the solution. Step : Substitute for x in one of the original equations. 3x y 6 First Equation. 3() y 6 Substitute for x. 6 y 6 Multiply. y 0 Subtract 6 from both sides. The solution to the system is (, 0). Step 3: Check the solution. Substitute the ordered pair into one of the original equations. If the solution is true for the first equation, then check the second equation. 5x y 10 5() (0) 10 Substitute (, 0) for x and y Simplify. 1. 3x y 10 3x y 10 3x y 10 The solution is (, 1). 3x y 1 3x y 1 3( ) y 10 Check: 3x y 1 6x x 1 y 10 3( ) ( 1) 1. x y 1 3. x y 11 x y 3 (, 1) y 1 1 y x 6y 11. 6x y 5. x 3y 6 x y x y (1, 1) (, 1) x y x y 6 x y 1 x y 8. A garden is feet longer than 3 times its width. If the perimeter of the garden is 6 feet, what is the length? (Hint: Use 3w and w to represent the length and width.) 5ft (11, 0) ( 3, ) ( 1, 1) Saxon. All rights reserved. 137 Saxon Algebra 1

8 continued 63 Solve the system of equations by elimination and check the solution. x 5y 3 x 3y 7 If the second equation is multiplied by, the equations will have equal and opposite coefficients for the variable x. Step 1: Multiply the second equation by. ( x 3y 7) x 6y 1 Step : Add the equations vertically. x 5y 3 x 6y 1 11y 11 Combine like terms. y 1 Divide by 11. The value of y in the solution is 1. Step 3: Substitute 1 for y in one of the original equations. x 5y 3 First Equation x 5( 1) 3 Substitute 1 for y. x 5 3 Multiply. x 8 Add 5 to both sides. x Divide both sides by. The solution to the system is (, 1). Step : Check the solution. Substitute the ordered pair into one of the original equations. x 3y 7 Second Equation () 3( 1) 7 Substitute (, 1) for x and y Simplify. Solve each system of equations by elimination. 9. 5x y 10 3(5x y 10) 15x 6y 30 3x 6y 66 3x 6y 66 3x 6y 66 18x 36 3( ) 6y 66 The solution is (, 10). x 6 6y 66 6y 60 y x y 3 (1, 1) 11. x y 0 x y 8 5x 3y (, ) 1. Amanda has dimes and quarters in her pocket totaling $.95. She has 16 coins in all. Write and solve a system of equations to find the number of quarters. 9 quarters Saxon. All rights reserved. 138 Saxon Algebra 1

9 Name Date Class You have learned to recognize and write examples of direct variation. Now you will learn to recognize and write examples of inverse variation. Direct variation: as the value of one variable increases, the value of the other variable also increases. Inverse variation: as the value of one variable increases, the value of the other variable decreases. Inverse variation can be represented by the equation y k x. The product of xy is always a constant, k. Is xy 0 an example of inverse variation? Solve the equation for y. xy x 0 x y 0 x Substitute a few values for x into the equation. When x, y When x, y 0 5. This is an inverse variation because as the value of x increases, the value of y decreases. Write an inverse variation relating x and y when y 15 and x 3. First, write the equation for inverse variation. y x k Substitute the values for x and y and solve for k. 15 k 3 k 5 Use this value for k in the equation for inverse variation. y 5 x Complete the steps to solve. 1. Is the relationship y x an inverse. Write an inverse variation relating x and y 8 variation? Explain. when y 8 and x. y 8 8 x y k 8 x y 8x 8 k This is not an inverse variation k 16 because as x increases, y also increases. Saxon. All rights reserved. 139 Saxon Algebra 1 y 16 x Solve. 3. Is the relationship xy an inverse. Write an inverse variation relating x and y variation? Explain. when y 0.5 and x 10. y x. This is an inverse variation because as x increases y decreases. Identifying, Writing, and Graphing Inverse Variation 6 y 5 x

10 continued 6 For the inverse variation, xy k, there can be several values for x and y as long as their product is equal to the constant k. For example, if k 0, x and y could be any values that make the equation xy 0 true. This is called the product rule for inverse variation. You can use this rule to solve for a missing value in an inverse variation. If y varies inversely as x and y 3 when x 3, find x when y. Use the product rule for inverse variation, x 1 y 1 x y. (3)(3) x () Substitute the values for x and y into the rule. 96 x Multiply. x Divide both sides by. When y, x. Complete the steps to solve. 5. If y varies inversely as x and y 6 when 6. If y varies inversely as x and y 8 Solve. x 18, find x when y 1. x 1 y 1 x y x 1 y 1 x y when x 3, find y when x. (18)(6) x 1 ( 3)(8) y x y x 9 y 6 When y 1, x 9. When x, y If y varies inversely as x and y 3 when 8. If y varies inversely as x and y 3 x 1, find x when y 6. when x 5, find y when x 15. When y 6, x 6. When x 15, y If y varies inversely as x and y If y varies inversely as x and y 0.5 when x 6, find y when x 10. when x 16, find x when y =. When x 10, y 3. When y, x. 11. Jan drives for 3 hours at an average rate of 50 miles per hour. How long would it take her to drive the same distance if she drove at an average rate of 60 miles per hour? Solve the following to find your answer: (50)(3) (60) y..5 hours 1. One rectangle has a width of 3 cm and a length of 1 cm. A second rectangle has the same area as the first, but has a width of cm. What is the length of the second rectangle? Solve the following to find your answer: (3)(1) () y. 9 cm Saxon. All rights reserved. 10 Saxon Algebra 1

11 Name Date Class Writing Equations of Parallel and Perpendicular Lines 65 You have learned to write the equation of a line using the slope and y-intercept. Now you will learn to write the equations of parallel and perpendicular lines. Write an equation in slope-intercept form for the line that passes through the point ( 3, ) and is parallel to y x 1. Use the point-slope form for the equation of a line. (y y 1 ) m(x x 1 ) Substitute the x- and y-values of the point given for x 1 and y 1, and the slope of the line for m. Use the slope of the line from the original equation to create an equation of a line parallel to it. x 1 3, y 1, and m 1 y 1 x 3 y x 3 Distributive Property y x 5 Add to both sides. Complete the steps to find the equation of the given line. 1. Write an equation in slope-intercept form for the line that passes through the point (, ) and is parallel to y 1 5 x 3 5. y y 1 m x x 1 y 1 5 x y 1 5 x 5 y 1 5 x 5 Find the equation of the given line.. Write an equation in slope-intercept form for the line that passes through the point (1, ) and is parallel to y x 3. y x 3. Write an equation in slope-intercept form for the line that passes through the point (1, 1) and is parallel to y 1 x. y 1 x 1 Saxon. All rights reserved. 11 Saxon Algebra 1

12 continued 65 Two lines are perpendicular if the slope of one line is the negative reciprocal of the slope of the other. For example, if the slope of a line is, the slope of a line perpendicular to it is 3 3. Write an equation in slope-intercept form for the line that passes through the point ( 1, ) and is perpendicular to y x. Use the point-slope form for the equation of a line. (y y 1 ) m(x x 1 ) Substitute the x- and y-values of the point given for x 1 and y 1. This time, use the negative reciprocal of the slope given to replace m in the point-slope form. The negative reciprocal of is 1. x 1 1, y 1, and m 1 (y ) 1 (x 1) y 1 x 1 Distributive Property y 1 x 1 Subtract from both sides. Complete the steps to find the equation of the given line.. Write an equation in slope-intercept form for the line that passes through the point (, ) and is perpendicular to y 3x 3. (y y 1 ) m(x x 1 ) (y ) 1 3 y 1 3 x (x ) 3 y 1 3 x Find the equation of the given line. 5. Write an equation in slope-intercept form for the line that passes through the point (0, 1) and is perpendicular to y 1 6. Write an equation in slope-intercept form for the line that passes through the point (8, 5) and is perpendicular to y x 5. x 1. y x 1 y 1 x 9 Saxon. All rights reserved. 1 Saxon Algebra 1

13 Name Date Class Solving Inequalities by Adding or Subtracting 66 You have learned to solve equations by adding and subtracting. Now you will learn to solve inequalities by adding or subtracting. Solve the inequality x 8 3. Then graph and check the solution. x 8 3 _ 8 8 _ Add 8 to each side. x 5 Simplify. Now graph the solution on a number line The solution includes all values less than 5. Check to see if the inequality symbol is pointing in the correct direction. Choose a number less than 5 and substitute it for x in the inequality This inequality is true, so the direction of the inequality symbol is correct. 1. Solve the inequality x 1 1. Then graph the solution. x x 0 Add 1 to each side. Simplify Solve the inequality x Then graph the solution x 3. Solve the inequality x 0. Then graph the solution x Saxon. All rights reserved. 13 Saxon Algebra 1

14 You can also subtract the same number from both sides of an inequality to solve it. continued 66 Solve the inequality x Then graph the solution. x 1 15 _ 1 1 _ Subtract 1 from each side. x 3 Simplify. Now graph the solution on a number line The solution includes all values greater than or equal to 3. Check to see if the inequality symbol is pointing in the correct direction. Choose a number greater than 3 and substitute it for x in the inequality The inequality is true, so the direction of the inequality symbol is correct.. Solve the inequality x. Then graph the solution. x x 6 Subtract from each side. Simplify Solve the inequality x Then graph and check the solution. Sample: x 5; 16 1, Aidan wants to buy a skateboard that will cost at least $75. He has already saved $7. Write and solve an inequality to find out how much money Aidan still has to save. x 7 75; x 8. Aidan needs to save at least $8. Saxon. All rights reserved. 1 Saxon Algebra 1

15 Name Date Class Solving and Classifying Special Systems of Linear Equations 67 You have learned to solve systems of linear equations. Now you will learn to recognize and solve special systems of linear equations. Solve the system of equations. 3x y 3x y 3 x y 1 y 3 x 1 Write the equations in slope-intercept form. 3 x y 1 y 3 x 1 The equations are identical. They will produce the same line, with an infinite number of solutions. A system of equations that has an infinite number of solutions is called consistent and dependent. Solve the system of equations. 3x y 5 3x y 7 y 3x 5 y 3x 7 Write the equations in slope-intercept form. Both equations have the same slope, 3, but they have different y -intercepts. This means these equations form parallel lines. Since parallel lines never intersect, the system has no solution. A system of equations that produces parallel lines with no solution is called inconsistent. Solve each system of equations. Classify each system as either consistent and dependent or inconsistent. 1. 6x y 9 3x y 1. x 3y 5 6x 9y 15 Write the equations in slope-intercept form. Write the equations in slope-intercept form. 6x y 9 y 3x 1 3x y 1 y 3x 1 There is no solution(s). The system is inconsistent. x 3y 5 y 3 x 5 3 6x 9y 15 y 3 x 5 3 There is an infinite number of solution(s). The system is consistent and dependent. 3. x y 6 y 6 x. 5x y 10x y y x 6 y x 6 infinite number of solutions; consistent and dependent y 5x y 5x no solutions; inconsistent Saxon. All rights reserved. 15 Saxon Algebra 1

16 continued 67 Solve the system of equations. x y 5 x y 0 x y 5 y x 5 Step 1: Write the equations in slope-intercept form. x y 0 y 1 x The slopes are not the same. x 1 x 5 Step : Substitute y 1 x into the first equation. 1 x 5 Remember that the expression for x or y obtained from an equation must never be substituted back x into the same equation. y 0 Step 3: Substitute for x in the second equation. y y 1 The solution is (, 1). The lines of these two equations intersect at only one point (, 1). A system of equations that has exactly one solution is called consistent and independent. Solve each system of equations. Classify each system as consistent and dependent, consistent and independent, or inconsistent. Show your work. 5. y x 3 6. y x y x 1 y 6. (, 1); consistent and independent (, 6); consistent and independent y x 3 y x 3 y x 1 y x 1 x 1 x 3 x 1 3 x y ( ) 1 7. Karen is 6 years older than her brother Ben. In years, Karen will be twice as old as Ben. How old are Karen and Ben? Use this system of equations to solve the problem. x (y ) x y 6 Karen is 10 and Ben is. Saxon. All rights reserved. 16 Saxon Algebra 1

17 Name Date Class Mutually Exclusive and Inclusive Events 68 When two events cannot happen at the same time, they are called mutually exclusive. For example, when rolling a number cube, the events rolling a 6 and rolling an odd number are mutually exclusive because 6 is not an odd number. To find the probability of one or the other events occurring, add the probabilities of each. Probability of Mutually Exclusive Events If A and B are mutually exclusive events, then P(A or B) P(A) + P(B). Marilyn spins the spinner shown. What is the probability that it lands on a vowel or on an odd number? 1 A The spinner is divided into 6 sections. One section is a vowel, A, and two sections are odd numbers, 1 and 3. Spinning a vowel and spinning an odd number are mutually exclusive. C 3 B P(A or B) P(A) + P(B) P(vowel or odd) = P(vowel) + P(odd) The probability of spinning a vowel or an odd number is 1. Complete the steps to find the probability of the events using the spinner above. 1. landing on B or on a number. landing on a number less than 3 or on C P(B or number) P(B) P(number) A marble is randomly chosen from a bag containing blue, 3 red, yellow, and 5 green marbles. Find the probability of the events. 3. pick a red marble or a green marble 3 P(less than 3 or C) P(less than 3) P(C) pick a yellow, blue, or green marble 3 Saxon. All rights reserved. 17 Saxon Algebra 1

18 When two events can occur at the same time, they are called inclusive, or overlapping. If you roll a number cube, rolling an odd number and rolling a prime number are inclusive events because 3 and 5 are both odd and prime. To find the probability of inclusive events, find the sum of the probabilities of each, and subtract the probability when both events occur. Probability of Inclusive Events If A and B are inclusive events, then P(A or B) P(A) P(B) P(A and B). Find the probability of rolling an odd number or a prime number on a number cube. outcomes that are odd: 1, 3, 5 P(odd) outcomes that are prime:, 3, 5 P(prime) outcomes that are odd and prime: 3, 5 P(odd and prime) 6 P(odd or prime) P(odd) P(prime) P(odd and prime) The probability that the number rolled is odd or prime is 3. continued 68 Saxon. All rights reserved. 18 Saxon Algebra Complete the steps to find the probability of the events when rolling a number cube labeled a number less than or an even number less than : 1,, 3 even:,, 6 less than and even: P(less than ) 1 P(even) 1 P(less than and even) 1 6 P(less than or even) P (less than ) P (even) P (less than and even) A letter is chosen from the word MATHEMATICS. Find the probability of the events. 6. choose a vowel or one of the last 5 letters 7. choose an M or one of the first 5 letters A deck of 0 cards has 5 red cards, 5 blue cards, 5 green cards and 5 yellow cards. The cards in each color group are numbered 1 through 5. If a card is chosen at random, what is the probability that it is red or a five? 5 11

19 Name Date Class Adding and Subtracting Radical Expressions 69 You have added and subtracted polynomials. Now you will add and subtract radical expressions. You can add and subtract radical expressions just like you add and subtract expressions with variables. _ x _ x 6x _ x _ y These are like terms. Add. _ 7 _ These are unlike terms. Do not add. _ 5 _ 3 These are like radicals. Add. These are unlike radicals. Do not add. Add These are like radicals (8 5) 10 Combine coefficients. Add ac 8 3ac. These are unlike radicals. Do not add. Subtract 10 7x 1 7x. These are like radicals. 10 7x 1 7x (10 1) 7x. Combine 7x coefficients. Subtract These are unlike radicals. Do not subtract. Complete the steps to add or subtract. All variables represent non-negative real numbers. 1. Add 3 y 8 y. These are like radicals. 3 y 8 y ( 3 8 ) y Combine coefficients. 11 y. Subtract 5 5. These are unlike radicals. Do not subtract. Add or subtract. All variables represent non-negative real numbers m 3m 10 3m Find the perimeter of a right triangle if the lengths of the two legs are 3 centimeters and centimeters and the hypotenuse is 5 centimeters. 1 cm Saxon. All rights reserved. 19 Saxon Algebra 1

20 continued 69 Sometimes it is necessary to simplify radical expressions before adding or subtracting. Simplify Simplify a 80 5 a Factor the radicands using perfect squares. 5 9 Product of Radicals Rule 5 3 Simplify. (5 3) Factor out. 8 a 80 5 a Combine like radicals. a a Factor the radicands using perfect squares. a a Product of Radicals Rule a 5 3a 5 Simplify. (a 3a) 5 Factor out 5. a 5 Simplify. Simplify. All variables represent non-negative real numbers Factor the radicands using perfect squares Product of Radicals Rule Simplify. ( 10 ) 3 Factor out Simplify p p 8 3p h h 7 h g g + 36 g g 6 6g 1. A rectangular mirror is 8 z inches wide and 7 z inches tall. What is its perimeter? 1z 3 in. Saxon. All rights reserved. 150 Saxon Algebra 1

21 Name Date Class Solving Inequalities by Multiplying or Dividing 70 You have solved inequalities by adding or subtracting. Now you will solve inequalities by multiplying or dividing. To solve an inequality involving multiplication or division, you can multiply or divide each side by the same number to isolate the variable. When the number you multiply or divide by is positive, the direction of the inequality symbol does not change. Solve and graph the inequality. 1 x 3 () 1 x 3() Multiply both sides by. x 1 Simplify. To graph the solution, draw a number line. Use a closed dot at 1 and shade to the right. Solve and graph the inequality. 5x 10 5x 10 Divide both sides by x Simplify. To graph the solution, draw a number line. Use an open dot at and shade to the left Complete the steps to solve and graph the inequality. 1. x 3 ( 3 ) x 3 ( 3 ) x. x x x Solve and graph each inequality x 7. 5 x 8 x 1 0 x x x 1 10 x 60 x Saxon. All rights reserved. 151 Saxon Algebra 1

22 When you solve an inequality, if the number you multiply or divide by is negative, you need to switch the direction of the inequality symbol. Solve and graph the inequality. x 1 ( 1) x ( 1) Multiply by 1 and switch 1 the symbol direction. x 8 Simplify. To graph the solution, draw a number line. Use an open dot at 8 and shade to the left. Solve and graph the inequality. 18 6x 18 6x Divide by 6 and switch 6 6 the symbol direction. 3 x continued 70 Simplify. To graph the solution, draw a number line. Use a closed dot at 3 and shade to the right Complete the steps to solve and graph the inequality. 7. x 8. 5x 5 9 ( 9 ) x 9 ( 9 ) 5x x 18 x Solve and graph each inequality x x 5 7 x 10 x x 1. 3x 1 x 3 x Marvin earns $15 for each lawn he mows. He is trying to earn at least $10 for a trip. Write and solve an inequality to find the number of lawns he needs to mow to earn enough money. Let x equal the number of lawns. 15x 10; at least 8 lawns Saxon. All rights reserved. 15 Saxon Algebra 1