Solving Inequalities
Inequalities and Their Graphs Objective: To write, graph, and identify solutions of inequalities.
Objectives I can write an inequality. I can identify solutions by evaluating inequalities. I can graph an inequality. I can write an inequality from a graph. I can write real-world inequalities.
Vocabulary An inequality is a mathematical sentence that uses an inequality symbol to compare the values of two expressions. You can use a number line to visually represent the values that satisfy an inequality
Writing Inequalities What inequality represent the verbal expression? All real numbers x less than or equal to -7 6 less than a number k is greater than 13 All real numbers p greater than or equal to 1.5 The sum of t and 7 is less than -3
Practice Write an inequality that represents each verbal expression. v is greater than or equal to 5 3 less than g is less than or equal to 17 B is less than 4 The quotient of k and 9 is greater than 1 3
Vocabulary A solution of an inequality is any number that makes the inequality true. The solutions of the inequality x < 5 are all real numbers that are less than 5. You can evaluate an expression to determine whether a value is a solution of an inequality.
Identifying Solutions by Evaluating Is the number a solution of 2x + 1 > 3? 3 1 Consider the numbers 0 1, 0, 1, and 3. Which are solutions of 13 7y 6?
Practice Determine whether each number is a solution of the given inequality. 1. 3y 8 > 22 2, 0, 5 2. 8m 6 10 2, 3, 1 3. 4x + 2 < 6 0, 2, 1 4. m(m 3) < 54 10, 0, 9 5. 6 n n 11 0.5, 2, 4
Vocabulary You can use a graph to indicate all of the solutions of an inequality. The open dot shows that 3 is not a solution. The closed dot shows that 3 is a solution. When the variable is larger than the number then you shade to the right. When the variable is smaller than the number then you shade to the left.
Graphing an Inequality What is the graph of each? 1. 2 a 2. x > 4 3. c < 0 4. 3 n 5. 1.5 < r
Practice Match the inequality with its graph. 1. x < 1 2. x 1 3. 1 < x 4. 1 x
Practice Graph each inequality. 1. y > 2 2. 3 < f 3. t 4 4. r 3 5 5. > p 4
Writing an Inequality From a Graph What inequality represents the graph? 0 0 0 0 0
Practice Write an inequality for each graph. 0 0 5 3 1 0 0 2
Writing Real World Inequalities What inequality describes the situation? Be sure to define a variable. 1. Trail Rides starting at $19.99 2. Speed Limit 8 miles per hour 3. The restaurant can set at most 172 people. 4. A person must be at least 35 years old to be elected President of the United States. 5. A light bulb can be no more than 75 watts to be safely used in this light fixture. 6. At least 475 students attended the orchestra concert Thursday night. 7. A law clerk has earned more than $20,000 since being hired.
Solving Inequalities Using Addition or Subtraction Objective: To use addition or subtraction to solve inequalities.
Objectives I can use the addition property of inequality. I can solve an inequality and check solutions. I can use the subtraction property of inequality. I can write and solve an inequality.
Vocabulary Just as you used the properties of equality to solve equations, you can use the properties of equality to solve inequalities. Equivalent inequalities are inequalities that have the same solutions. Addition Property of Inequality: Let a, b, and c be real numbers If a > b, then a + c > b + c If a < b, then a + c < b + c This property is also true for and. Example: 5 > 4, so 5 + 3 > 4 + 3 (8 > 7) 3 < 7, so 3 + 8 < 7 + 8 (11 < 15)
Using the Addition Property of Inequality What are the solutions? Graph the solution. 1. x 15 > 12 2. n 5 < 3 3. p 4 < 1 4. 8 d 2
Practice Solve each inequality. Graph and check your solutions. 1. y 2 > 11 2. t 4 7 3. v 4 < 3 4. s 10 1 5. 6 > c 2 6. 9 < p 3 7. 8 f 4 8. 3 x 1
Vocabulary The original inequality has infinitely many solutions, so you cannot check them all. However, you can verify that the final inequality is correct by checking its endpoint and the direction of the inequality symbol.
Solving an Inequality and Checking Solutions What are the solutions? 1. 10 x 3 2. m 11 2 3. z 12 4 4. 0 < 1 3 + f
Practice Solve each inequality. Graph and check your solutions. 1. 4.3 > 0.4 + s 2. 2.5 > n 0.9 3. 3 4 > r 3 4 4. c 4 7 < 6 7 5. y 1 1.5 6. p 1 1 2 > 1 1 2 7. 1 < d 7 8. a 3.3 2.6
Vocabulary Subtraction Property of Inequality: Let a, b, and c be real numbers If a > b, then a c > b c If a < b, then a c < b c This is also true for and. Example: 3 > -4, so 3 3 > -4 3 (0 > -7) -3 < 5, so -3 2 < 5 2 (-5 < 3)
Using the Subtraction Property of Inequality What are the solutions? 1. t + 6 > 4 2. 1 y + 12 3. y + 5 < 7 4. 4 + c > 7
Practice Solve each inequality. Graph and check your solutions. 1. x + 5 10 2. n + 6 > 2 3. 2 < 9 + c 4. 8.6 + z 14 5. 1 5 + b 6. 7. 8. 1 + a > 3 4 4 1 < n + 3 3 3 5 + d 2 5
Writing and Solving an Inequality The hard drive on your computer has a capacity of 120 GB. You have used 85 GB. You want to save some home videos to your hard drive. What are the possible sizes of the home video collection you can save? A club has a goal to sell at least 25 plants for a fundraiser. Club members sell 8 plants on Wednesday and 9 plants on Thursday. What are the possible numbers of plants the club can sell on Friday to meet their goal?
Practice Your goal is to take at least 10,000 steps per day. According to your pedometer, you have walked 5274 steps. Write and solve an inequality to find the possible number of steps you can take to reach your goal. You earn $250 per month from your part time job. You are in a kayaking club that costs $20 per month, and you save at least $100 each month. Write and solve an inequality to find the possible amounts you have left to spend each month.
Solving Inequalities Using Multiplication or Division Objective: to use multiplication or division to solve inequalities.
Objectives I can multiply by a positive number to solve an inequality. I can multiply by a negative number to solve an inequality. I can divide by a positive number to solve an inequality. I can divide by a negative number to solve an inequality.
Vocabulary Just as you used multiplication and division to solve equations, you use multiplication and division to solve inequalities.
Multiplication Property Let a, b, and c be real numbers with c > 0. If a > b, then ac > bc. If a < b, then ac < bc. Example: 4 > 1, so 4(3) > 1(3) (12 > 3) Example: -2 < 3, so -2(2) < 3(2) (-4 < 6) Let a, b, and c be real numbers with c < 0. If a > b, then ac < bc. If a < b, then ac > bc. Example: 3 > 1, so 3(-1) < 1(-1) (-3 < -1) Example: 2 < 4, so 2(-2) > 4(-2) (-4 > -8)
Vocabulary Here s Why It Works: Multiplying or dividing each side of an inequality by a negative number changes the meaning of an inequality. You need to reverse the inequality symbol to make the inequality true. Here is an example: 3 > 1 2(3) 2(1) Multiply by 2 6 2 Simplify 6 < 2 Reverse the inequality symbol to make the inequality true
Multiplying by a Positive Number What are the solutions? 1. 2. 3. x 3 < 2 c 8 > 1 4 x 3 1 4. 1 b 8
Practice Solve each inequality. Graph and check your solution. 1. 2. x 2 5 w < 1 6 3. 4 > p 8 4. 3 < x 3 5. 7 7 3 x 6. 8 > 2 3 k
Multiplying by a Negative Number What are the solutions? 1. 3 w 3 4 2. n < 1 3 3. x > 1 5 4. 1 x 9
Practice Solve each inequality. Graph and check your solution. 1. 1 5 y 4 2. v 1.5 2 3. 0 3 m 11 4. 3 b < 6 2 5. 3 > 3 m 4 8 6. 5 5 y 9
Vocabulary Solving inequalities using division is similar to solving inequalities using multiplication. If you divide each side of an inequality by a negative number, you need to reverse the direction of the inequality symbol. When dividing by a negative number, remember to change your sign! It is incorrect if you do not change your sign.
Division Property Let a, b, and c be real numbers with c > 0. If a > b, then a/c > b/c. If a < b, then a/c < b/c. Example: 6 > 3, so 6/3 > 3/3 (2 > 1) Example: 8 < 12, so 8/4 < 12/4 (2 < 3) Let a, b, and c be real numbers with c < 0. If a > b, then a/-c < b/-c. If a < b, then a/-c > b/-c. Example: 6 > 3, so 6/-3 < 3/-3 (-2 < -1) Example: 8 < 12, so 8/-4 > 12/-4 (-2 > -3)
Dividing by a Positive Number You walk dogs in your neighborhood after school. You earn $4.50 per dog. How many dogs do you need to walk to earn at least $75? A student club plans to buy food for a soup kitchen. A case of vegetables costs $10.68. The club can spend at most $50 for this project. What are the possible numbers of cases the club can buy?
Practice Solve each inequality. Graph and check your solution. 1. 3m 6 2. 4t < 12 3. 11z > 33 4. 18b 3 5. 8t 64 6. 63 7q
Dividing by a Negative Number What are the solutions? 1. 9y 63 2. 5x > 10 3. 3x 9 4. 6x < 36
Practice Solve each inequality. Graph and check your solution. 1. 30 > 5c 2. 4w 20 3. 56 < 7d 4. 7y 17 5. 5h < 65 6. 12x > 132
Practice Text messages cost $0.15 each. You can spend no more than $10. How many text messages can you send? Tetras cost $3.99 each. You can spend at most $25. How many tetras can you buy for your aquarium?
More Algebraic Properties
Vocabulary The following properties can help you understand algebraic relationships. Properties of Equality Examples Reflexive Property a = a 5x = 5x, $1 = $1 Symmetric Property If a = b, then b = a If 15 = 3t, then 3t = 15. If 1 pair = 2 socks, then 2 socks = 1 pair. Transitive Property If d = 3y and 3y = 6, then d = 6. If 36 in = 3 ft and 3 ft = 1 yd, then 36 in = 1 yd Properties of Inequality Examples Transitive Property If 8x < 7 and 7 < y, then 8x < y. If 1 cup < 1 qt and 1 qt < 1 gal, then 1 cup < gal.
Practice Complete the statement using the given property. If 7x < y and y < x + 2, then 7x <. (Transitive Property of Inequality) If 2000 lb = 1 ton, then 1 ton =. (Symmetric Property) Name the property. If 3.8 = n, then n = 3.8. 6 in = 6 in If x = 7 and 7 = 5 + 2, then x = 5 + 2 If math class is earlier than art class and art class is earlier than history class, then math class is earlier than history class. Complete the following sentence. If Amy is shorter than Greg is shorter than Lisa, then Amy is shorter than.
Solving Multi-Step Inequalities Objective: To solve multi-step inequalities.
Objectives I can solve inequalities using more than one step. I can write and solve a multi-step inequality. I can solve inequalities using the distributive property. I can solve an inequality with variables on both sides. I can solve inequalities with special solutions.
Vocabulary You solve a multi-step inequality in the same way you solve a onestep inequality. You use the properties of inequality to transform the original inequality into a series of simpler, equivalent inequalities.
Using More than One Step What is the solution? 1. 9 + 4t > 21 2. 6a 7 17 3. 4 < 5 3n 4. 50 > 0.8x + 30
Practice Solve each inequality. Check your solutions. 1. 5f + 7 22 2. 6 3p 9 3. 6n 3 > 18 4. 9 12 + 6r 5. 5y 2 < 8 6. 6 12 + 4j
Vocabulary You can adapt familiar formulas to write inequalities. You use the realworld situation to determine which inequality symbol to use.
Writing and Solving a Multi-Step Inequality In a community garden, you want to fence in a vegetable garden that is adjacent to your friend s garden. You have at most 42 ft of fence. What are the possible lengths of your garden? 1 2 F E E T Your Garden
Writing and Solving a Multi-Step Inequality You want to make a rectangular banner that is 18 feet long. You have no more than 48 ft of trim for the banner. What are the possible width of the banner?
Practice On a trip from Buffalo, New York, to St. Augustine, Florida, a family wants to travel at least 250 mi in the first 5 hours of driving. What should their average speed be in order to meet this goal? An isosceles triangle has at least two congruent sides. The perimeter of a certain isosceles triangle is at most 12 in. The length of each of the two congruent sides is 5 in. What are the possible lengths of the remaining side?
Using the Distributive Property What is the solution? 1. 3 t + 1 4t 5 2. 15 5 2 4m + 7 3. 2 p + 2 3p 1 4. 2(18x 5) 44
Practice Solve each inequality. 1. 3 k 5 + 9k 3 2. 7c 18 2c > 0 3. 3 j + 3 + 9j < 15 4. 4 4 6y 12 2y 5. 30 > 5z + 15 + 10z 6. 4 d + 5 3d > 8
Vocabulary Some inequalities have variables on both sides of the inequality symbol. You need to gather the variable terms on one side of the inequality and the constant terms on the other side.
Solving an Inequality With Variables on Both Sides What are the solutions? 1. 6n 1 > 3n + 8 2. 3b + 12 > 27 2b 3. 18x 5 3 6x 2 4. 4w 8 6w + 10
Practice Solve each inequality. 1. 4x + 3 < 3x + 6 2. 6 3p 4 p 3. 4v + 8 6v + 10 4. 3m 4 6m + 11 5. 5f + 8 2 + 6f 6. 4t + 17 > 7 + 5t
Vocabulary Sometimes solving an inequality gives a statement that is always true, such as 4 > 1. In that case, the solutions are all real numbers. If the statement is never true, as is 9 5, then the inequality has no solution.
Inequalities With Special Solutions What are the solutions? 1. 10 8a 2 5 4a 2. 6m 5 > 7m + 7 m 3. 9 + 5n 5n 1 4. 8 + 6x 7x + 2 x
Practice Solve each inequality, if possible. If the inequality has no solution, write no solution. If the solutions are all real numbers, write all real numbers. 1. 3 w 3 9 3w 2. 9 + 2x < 7 + 2 x 3 3. 5r + 6 5 r + 2 4. 2 n 8 < 16 + 2n 5. 2 6 + s 15 2s 6. 6w 4 2(3w + 6)
Compound Inequalities Objective: To solve and graph inequalities containing the word and. To solve and graph inequalities containing the word or.
Objectives I can write a compound inequality. I can solve a compound inequality involving the word AND. I can write and solve a compound inequality. I can solve a compound inequality involving the word OR. I can use interval notation.
Vocabulary A compound inequality consists of two distinct inequalities joined by the word and or the word or. You find the solutions of a compound inequality either by identifying where the solution sets of the distinct inequalities overlap or by combining the solution sets to form a larger solution set.
Vocabulary The graph of a compound inequality with the word and contains the overlap of the graphs of the two inequalities that form the compound inequality. You can rewrite a compound inequality involving and as a single inequality. For instance, x 2 and x 4. You read this as x is greater than or equal to 2 and less than or equal to 4. Another way to read it is x is between 2 and 4, inclusive. Inclusive means the solutions of the inequality include both 2 and 4. The graph a compound inequality with the word or contains each graph of the two inequalities that form the compound inequality.
Writing a Compound Inequality What compound inequality represents the phrase? Graph the inequality. 1. All real numbers that are greater than 2 and less than 6 2. All real numbers that are less than or equal to 2 1 or greater than 6 2 3. All real numbers that are less than 0 or greater than or equal to 5 4. All real numbers that are greater than or equal to 4 and less than 8
Practice Write a compound inequality that represents each phrase. Graph the solutions. 1. The circumference of a women s basketball much be between 28.5 inches and 29 inches, inclusive. 2. All real numbers that are between 5 and 7 3. All real numbers that are greater than or equal to 0 and less than 8
Vocabulary A solution of a compound inequality involving and is any number that makes both inequalities true. One way you can solve a compound inequality is by separating it into two inequalities.
Solving a Compound Inequality Involving AND What are the solutions? Graph the solutions. 1. 3 m 4 < 1 2. 2 < 3y 4 < 14 3. 5 y + 2 11 4. 1 < 2x 7 4 2 < 5
Practice Solve each compound inequality. Graph your solutions. 1. 2 0.75w 4.5 2. 3 6 q 9 3 3. 4 < k + 3 < 8 4. 15 20+11+k 3 5. 3 < 4p 5 3 19 6. 4 3 1 7 w 3 4 < 1
Vocabulary You can also solve an inequality like 3 m 4 < 1 by working on all three parts of the inequality at the same time. You work to isolate the variable between the inequality symbols. This method is used in the next example.
Writing and Solving a Compound Inequality To earn a B in your algebra course, you must achieve an unrounded test average between 84 and 86, inclusive. You scored 86, 85, and 80 on the first three tests of the grading period. What possible scores can you earn on the fourth and final test to earn a B in the course? Suppose you scored 78, 78, and 79 on the first three tests. Is it possible for you to earn a B in the course? Assume that 100 is the maximum grade you can earn in the course and on the test.
Practice Your test scores in science are 83 and 87. What possible scores can you earn on your next test to have a test average between 85 and 90, inclusive? For safety, the weight of each rider of a certain roller coaster must fall in the range given by the inequality 10 w 30 70 where w is in 2 pounds. Solve for w to find the safe weight range.
Vocabulary A solution of a compound inequality involving or is any number that makes either inequality true. To solve a compound inequality involving or, you must solve separately the two inequalities that form the compound inequality.
Solving a Compound Inequality Involving OR What are the solutions of each? Graph the solutions. 1. 3t + 2 < 7 or 4t + 5 < 1 2. 2y + 7 < 1 or 4y + 3 5 3. 6b 1 < 7 or 2b + 1 > 5 4. 4d + 5 13 or 7d 2 < 12
Practice Solve each compound inequality. Graph your solutions. 1. 5y + 7 3 or 3y 2 13 2. 5 + m > 4 or 7m < 35 3. 7 c < 1 or 4c 12 4. 5z 3 > 7 or 4z 6 < 10 5. 7x > 42 or 5x 10
Vocabulary You can use an inequality such as x 3 to describe a portion of the number line called an interval. You can also use interval notation to describe an interval on the number line. Interval notation includes the use of three special symbols. These symbols include: Parentheses: Brackets: Infinity: Use ( or ) when a < or > symbol indicates that the interval s endpoints are NOT included. Use [ or ] when a or symbol indicates that the interval s endpoints ARE included. Use when the interval continues forever in a positive direction. Use when the interval continues forever in a negative direction. x > 2 (2, ) 3 x 12 [3,12] x < 2 or x 3 (,2) or [3, )
Using Interval Notation What is the inequality? Graph each inequality. 1. [ 4,6) 2. (, 1] or (2, ) 3. ( 2,7] What is the interval notation? Graph the interval. 1. x 6 2. c < 8 3. 5 b <15
Examples Write each interval as an inequality. 1. (, -1] or (3, ) 2. [6, ) 3. [ 4,5] 4. (,2] Write each inequality in interval notation. 1. x > 2 2. x 0 3. x < 2 or x 1 4. 3 x < 4
Absolute Value Equations and Inequalities Objective: To solve equations and inequalities involving absolute value.
Objectives I can solve an absolute value equation. I can solve an absolute value equation using real world examples. I can solve an absolute value equation with no solution. I can solve an absolute value inequality involving. I can solve an absolute value inequality involving.
Vocabulary You can solve absolute value equations and inequalities by first isolating the absolute value expression, if necessary. Then write an equivalent pair of linear equations or inequalities.
Solving an Absolute Value Equation What are the solutions? 1. x + 2 = 9 2. n 5 = 2 3. b = 1 2 4. y = 4 5. f 6 = 10
Practice Solve each equation. 1. x 10 = 2 2. 5 d = 20 3. n + 3 = 7 4. 3 m = 9 5. y + 3 = 3 6. 7 = s 3
Vocabulary Some equations, such as 2x 5 = 13, have variable expressions within absolute value symbols. The equation 2x 5 = 13 means that the distance on a number line from 2x 5 to 0 is 13 units. There are two points that are 13 units from 0: 13 and 13. So you find the values of x, solve the equation 2x 5 = 13 and 2x 5 = 13. You can generalize this process as follows. Solving Absolute Value Equations: To solve an equation in the form A = b, where A represents a variable expression and b > 0, solve A = b and A = b.
Solving an Absolute Value Equation Starting from 100 ft away, your friend skates toward you and then passes by you. She skates at a constant speed of 20 ft/s. Her distance d from you in feet after t seconds is given by d = 100 20t. At what time is she 40 feet from you? Another friend s distance d from you (in feet) after t seconds is given by d = 80 5t. What does the 80 in the equation represent? What does the 5 in the equation represent? At what time is she 60 feet from you?
Practice Solve each equation. If there is no solution, write no solution. 1. 2x 5 = 13 2. r 8 = 5 3. c + 4 = 6 4. 2 = g + 3 5. 3 = m + 2 6. 2t = 6
Vocabulary Recall that absolute value represents distance from 0 on a number line. Distance is always nonnegative. So any equation that states the absolute value of an expression is negative has no solutions.
Solving an Absolute Value Equation With No Solution What are the solutions? 1. 3 2z + 9 + 12 = 10 2. 3x 6 5 = 7 3. 2 7d = 14 4. 3 2w = 12 5. 3 v 3 = 9 6. 2 d + 4 = 8
Practice Solve each equation. If there is no solution, write no solution. 1. 4f + 1 2 = 5 2. 3t 2 + 6 = 2 3. 4 2y 3 1 = 11 4. 3 x + 2 + 4 = 13 5. 4 k = 12 6. 3n 2 = 4 7. 4 k + 1 = 16
Vocabulary You can write absolute value inequalities as compound inequalities. The graphs below show two absolute value inequalities. x > 2 represents all numbers with a distance from 0 that is greater than 2 units. So x > 2 means x < 2 or x > 2. This is also true with the inequality symbol. x < 2 represents all numbers with a distance from 0 that is 2 units. So x < 2 means 2 < x < 2. This is also true with the inequality symbol.
Vocabulary Solving Absolute Value Inequalities: To solve an inequality in the form A < b, where A is a variable expression and b > 0, solve the compound inequality b < A < b. (THIS IS AN AND INEQUALITY!) To solve an inequality in the form A > b, where A is a variable expression and b > 0, solve the compound inequality A < b or A > b. (THIS IS AN OR INEQUALITY!) Similar rules are true for ΙAΙ b and ΙAΙ b.
Solving an Absolute Value Inequality Involving What are the solutions? Graph the solutions. 1. x 3 2. y + 8 3 3. 4w + 1 > 11 4. 5t 4 16 5. x + 2 > 1 6. 2 9c 13 > 20
Practice Solve and Graph each inequality. 1. 3t + 1 > 8 2. 4x + 7 > 19 3. 5m 9 24 4. 8n 24 5. 4 3 m + 2 > 14 6. 2x + 4 5
Solving an Absolute Value Inequality Involving A company makes boxes of crackers that should weigh 213 grams. A qualitycontrol inspector randomly selects boxes to weigh. Any box that varies from the weight by more than 5 grams is sent back. What is the range of allowable weights for a box of crackers? A food manufacturer makes 32 ounce boxes of pasta. Not every box weighs exactly 32 ounces. The allowable difference from the ideal weight is at most 0.05 ounces. Write and solve an absolute value inequality to find the range of allowable weights.
Practice Solve and graph each inequality. 1. x < 5 2. x + 3 < 5 3. y 2 1 4. p 7 3 5. 2c 5 < 9 6. 2f + 9 13 7. 2v 1 9 8. h 3 < 5