Algebra I Notes Unit Five: Linear Inequalities in One Variable and Absolute Value Equations & Inequalities

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1 Syllabus Objective 4.4 The student will solve linear inequalities and represent the solution graphically on a number line and algebraically. Inequality Symbols: < less than less than or equal to > greater than greater than or equal to Graph of a Linear Inequality: the set of points on the number line that represents all solutions of the inequality Ex: Graph the following inequalities on a number line. 1. x < 3 Solution: Graph (shade) all numbers that are less than 3. (Note: This does not include the number 3, therefore we will use an open circle on 3 on the number line.). x 4 Solution: Graph (shade) all numbers that are greater than or equal to 4. (Note: This does include the number 4, therefore we will use a closed circle on 4 on the number line.) 3. 1 < x Solution: We can rewrite this inequality as x > 1. Graph (shade) all numbers that are greater than 1. (Note: This does not include the number 1, therefore we will use an open circle on 1 on the number line.) 4. 1 x Solution: Graph (shade) all numbers that are less than or equal to 1. (Note: This does include the number 1, therefore we will use a closed circle on 1 on the number line.) The table below gives a summary of when to use an open or closed circle on the number line: Symbol < & > < & Boundary on a Number Line Open dot Does not include the point Closed (solid) dot Includes the point Page 1 of 15 McDougal Littell:

2 Writing an Inequality Ex: Write an inequality for each of the following. 1. Jay needs to score at least an 84% on his exam to earn an A for the course. Solution: Let Jay s score = J J 84. Misa can spend at most $15 on a new cell phone. Solution: Let Misa s amount = M M In order to win the tournament, River must beat the high bowling score of 5. Solution: Let River s score = R R > 5 4. The maximum number of students per class is 35. Solution: Let the number of students = s s 35 Solving a Linear Inequality: To solve a linear inequality, we will follow the same steps as solving a linear equation. However, we must be cautious about the direction of the inequality symbol! Explore the effects of transformations on the inequalities: Adding a Positive Constant 3 9 Add to both sides Inequality is still true. Adding a Negative Constant 3 9 Add to both sides ( ) ( ) Multiplying/Dividing by a Positive Constant Inequality is still true. 3 9 Multiply both sides by 1 3 (Note: This is the same as dividing both sides by 3.) Inequality is still true. 3 3 Multiplying/Dividing by a Negative Constant 3 9 Multiply both sides by 1 (Note: This is the same as dividing both sides by 3.) Inequality is false. ( 1 is NOT less than or equal to 3.) 3 3 Page of 15 McDougal Littell:

3 Summary: The process for solving a linear inequality is the same as solving a linear equation with one exception If you multiply or divide by a NEGATIVE constant when solving, you must REVERSE (FLIP) the inequality sign to make it true. Ex: Solve the linear inequality x >. Divide both sides by 1 to isolate the variable. x < 1 1 Since we divided by a NEGATIVE number, we must FLIP the inequality symbol. x < Note: The inequality above could have also been solved by switching sides of the inequality (change sides, change signs) to obtain > x, which is equivalent to x <. b Ex: Solve the linear inequality 3 and graph the solution on a number line. b Multiply both sides by to isolate the variable. 3 b 6 Note: We did not multiply by a NEGATIVE number, so there is no need to flip the inequality symbol. Graph: Ex: Solve the linear inequality 4< x + 5and graph the solution on a number line. Subtract 5 from both sides to isolate the variable. 4< x < x Rewrite the inequality with the variable on the left. (Note: This is not necessary, but it makes it easier to graph the solution.) x > 1 Graph: Ex: Solve the linear inequality 4 n. We will solve this inequality two different ways. Option 1: Subtract 4 from both sides to isolate the variable term. 4 n n Divide both sides by 1 to isolate the variable. 6 n 1 1 Note: FLIP! 6 n or n 6 Page 3 of 15 McDougal Littell:

4 Option : Bring the variable to the other side. 4 n + n + n n 4 Add to both sides to isolate the variable. n n 6 (Same as Option 1) You Try: Solve the inequalities and graph the solutions on a number line. n < p. 3. q 3 QOD: When is it necessary to flip the inequality symbol when solving a linear inequality? Explain why and give an example to support your explanation. Page 4 of 15 McDougal Littell:

5 Syllabus Objective 4.4 The student will solve linear inequalities and represent the solution graphically on a number line and algebraically. Solving Multi-Step Linear Inequalities Recall: The process for solving a linear inequality is the same as solving a linear equation with one exception If you multiply or divide by a NEGATIVE constant when solving, you must FLIP the inequality sign to make it true. Ex: Solve the inequality 8x + 6< 10 and graph the solutions on a number line. Step One: Undo addition (using subtraction). Step Two: Undo multiplication (using division). Step Three: Graph your solution. 8x + 6< x < 4 8x 4 > x > Note: FLIP! Ex: Solve the inequality x and graph the solutions on a number line. Step One: Isolate the variable term x x 6 3x Step Three: Undo multiplication (using division). 3 3 x or x Step Four: Graph your solution. Page 5 of 15 McDougal Littell:

6 Ex: Solve the inequality: 0 < 5y 3( 4 y) Step One: Use the distributive property. 0 < 5y 1 + 3y Step Two: Combine like terms. 0 < 8y 1 Step Three: Undo subtraction (using addition). Step Four: Undo multiplication (using division). 0 < 8y < 8y 3 8y < < y or y > 4 Ex: Solve the inequality 3x 4( x+ 6) Step One: Use the distributive property. 3x 4x+ 4 Step Two: Bring all variables to the same side. Step Three: Undo subtraction (using addition). 3x 4x+ 4 4 x 4x 1x 4 1x x 6 Step Four: Undo multiplication (using division). 1x x 6 Note: FLIP! Alternate Method: In the example above, we collected the variables on the left side. We could have also collected the variables on the right side. ( x ) 3x x 4x + 4 3x 4x+ 4 3 x 3x x + 4 x x or x 6 Page 6 of 15 McDougal Littell:

7 Application Problems with Linear Inequalities Ex: A carnival charges $6 for admission and $1.50 for each ride. Jake has $30 to spend at the carnival. Write an inequality that represents the number of rides Jake can go on. Let the number of rides = r Cost for carnival = Admission # of rides C = r Because Jake has $30, he can spend any amount less than or equal to $ r 30 Solve the inequality for r r r 4 r You Try: 1. Solve the inequality and graph the solution on a number line. 4 < ( n ). At Cedar Point, it costs $140 for a season pass. It costs $30 to buy a single ticket into the park. Write an inequality that represents how many times you could visit the park before spending more than the cost of a season pass. n 3 QOD: List a key word you may find in a word problem for each of the four inequality symbols. Page 7 of 15 McDougal Littell:

8 Syllabus Objective 4.6 The student will solve compound inequalities and represent the solution graphically on a number line and algebraically. Compound Inequality: two inequalities connected by and or or Writing a Compound Inequality: Union ( or statement): When ordering a drink at a restaurant, I order Coke or Pepsi. This means either one is okay. Ex: Write a compound inequality for the verbal statement and graph on a number line: x is greater than 4 or less than. x > 4 or x< Note: Either one is okay. Intersection ( and statement): When ordering at a restaurant, I order a Coke and a cheeseburger. This means I would like both. Ex: Write a compound inequality for the verbal statement and graph on a number line: x is greater than or equal to 3 and less than 4. x 3 and x< 4 can also be written as 3 x < 4 Note: The solution is where the shaded parts overlap Note: An and statement can be combined and written as one inequality. An or statement cannot! Ex: A refrigerator is designed to work on an electric line carrying from 115 to 10 volts. Write a compound inequality for the number of volts (v) the refrigerator is designed for. Solution: v 115 and v 10, which can be written as 115 v 10 Solving a Compound Inequality with OR Ex: Solve the compound inequality and graph the solution on a number line. 6x 5< 7 or 5 8x + 1 Step One: Solve each inequality separately. 6x 5< 7 5 8x x 1 4 8x < x< 3 x Page 8 of 15 McDougal Littell:

9 Step Two: Write as a compound inequality. x< or x 3 Step Three: Graph the union of the inequalities on a number line. Solving a Compound Inequality with AND Ex: Solve the compound inequality 3x + 1 7and graph the solution on a number line. Step One: Isolate the variable between the two inequality symbols using inverse operations. Note: What you do to the middle, you must do to both sides! 3x x x Note: The solution is read x is greater than or equal to 1 and less than or equal to. Step Two: Graph the intersection of the inequalities on a number line. Reminder: Be careful when multiplying or dividing by a negative number. You must FLIP the inequality symbol(s). Ex: Solve the compound inequality 3< 1 x < 5and graph the solution on a number line. Step One: Isolate the variable between the two inequality symbols using inverse operations. 3< 1 x < x 6 > > FLIP! 1> x > 3 This is read x is less than 1 and greater than 3 Note: We will rewrite the inequality with the numbers in order to match the order of numbers on the number line. 3< x < 1 Step Two: Graph the intersection of the inequalities on a number line. Page 9 of 15 McDougal Littell:

10 Special Cases: Ex: Find the solutions to the compound inequality 4< x < 1. Solution: This is read x is greater than 4 AND less than 1. Because this is an and statement, we are looking for the intersection of these inequalities. Graphing each one separately, we have x > 4 x < 1 Looking at the graphs above, we see there is NO intersection! Therefore, there is NO SOLUTION to this compound inequality. Ex: Find the solutions to the compound inequality x< 3 or x > 0. Solution: This is read x is less than 3 OR greater than 0. Because this is an or statement, we are looking for the union of these inequalities. Graphing each one separately, we have You Try: x < 3 x > 0 Looking at the graphs above, the union (or overlap) of each inequality would be the entire number line! Therefore, the solution would be ALL REAL NUMBERS. 1. Write an inequality that represents the situation: Water is a non-liquid when the temperature is 3 F or below, or when the temperature is at least 1 F.. Write an inequality that represents the graph below. 3. Solve the compound inequality and graph the solution on a number line. 11 3x < 1 4. Solve the compound inequality and graph the solution on a number line. 5 < x 3 or 3x+ > 1 QOD: In your own words, describe the difference between an or and an and statement with respect to compound inequalities. Page 10 of 15 McDougal Littell:

11 Syllabus Objective: 4.5 The student will solve absolute value equations both algebraically and graphically on a number line. Absolute Value: the distance a number is away from the origin on the number line Algebraic definition of absolute value: a a, a 0 = a, a < 0 Ex: Solve the absolute value equation: x = 3 Note: This means that the distance from the origin is 3. Solution: There are two numbers that are 3 units away from the origin: x = 3 and x = 3 When solving an absolute value equation in the form ax + b = c ax + b = c and ax + b = c., you must solve the equations Ex: Solve the equation x + 8 = and graph the solution(s) on a number line. Step One: Rewrite as two equations. x+ 8= x+ 8= Step Two: Solve both equations. x+ 8= x+ 8= x = 6 x = 10 Step Three: Plot the points on a number line. On your own: Check your solutions by substituting them back into the original equation. Ex: Solve the absolute value equation: 5x 1 + 3= 14 Step One: Put the equation in ax + b = c form. 5x 1 = 11 Step Two: Rewrite as two equations. 5x 1= 11 5x 1= 11 Step Three: Solve both equations. 5x = 1 5x = 10 Solutions: 1 x = or x = 5 One your own: Check your solutions by substituting them back into the original equation. Page 11 of 15 McDougal Littell:

12 Special Cases: Absolute Value Equations with One Solution Ex: Solve the equation: 3n = Step One: Put the equation in ax + b = c form. 3n = 3n + 4 = 0 Step Two: Rewrite as two equations. 3n+ 4= 0 3n+ 4= 0 (Note: +0 and 0 both equal 0!) 3n + 4= 0 Step Three: Solve the equation. 3n + 4= n 4 = 3 3 n = 4 3 Absolute Value Equations with No Solution Ex: Solve the equation 3 x 8 = 6. Step One: Put the equation in ax + b = c form. 3 x 8 6 = 3 3 x 8 = Note: This reads that the absolute value of the quantity x minus 8 equals. This can never be true! The absolute value of any number or expression must be a non-negative number. Therefore, this equation has NO SOLUTION. You Try: Solve the equation 10 = x 4 8 and graph the solution(s) on a number line. QOD: Explain why an absolute value equation can have two solutions. Page 1 of 15 McDougal Littell:

13 Syllabus Objective 4.7 The student will solve absolute value inequalities and represent the solution graphically on a number line and algebraically. Solving Absolute Value Inequalities The inequality x < 4 means that we are looking for all numbers, x, whose distance from 0 on the number line is less than 4. Graphically, the solutions are: Algebraically, we would use the compound inequality 4< x < 4to represent the solutions to the absolute value inequality. The inequality x > 3 means that we are looking for all numbers, x, whose distance from 0 the number line is greater than 3. Graphically, the solutions are: Summary: Algebraically, we would use the compound inequality x< 3 or x > 3 to represent the solutions to the absolute value inequality. Inequality Solutions ax + b < c ax + b < c AND ax + b > c ax + b c ax + b c AND ax + b c ax + b > c ax + b > c OR ax + b < c ax + b c ax + b c OR ax + b c ax + b = c ax + b = c OR ax + b = c Ex: Solve the absolute value inequality x 5 and graph the solution on a number line. Step One: Rewrite as a two inequalities. x 5 AND x 5 Step Two: Solve each inequality. x 7 x 3 Step Three: Write the solution as a compound inequality. 3 x 7 Step Four: Graph the solution on a number line. Page 13 of 15 McDougal Littell:

14 Ex: Solve the absolute value inequality 3x 3 + 4> 10 and graph the solution on a number line. Step One: Rewrite the inequality in ax + b > c form. 3x 3 + 4> x 3 > 6 Step Two: Rewrite as two inequalities. 3x 3> 6 OR 3x 3< 6 Step Three: Solve each inequality. 3x > 9 x > 3 3x < 3 x < 1 Step Four: Write the solution as a compound inequality. x< 1 or x> 3 Step Five: Graph the solution on a number line. Ex: Solve the absolute value inequality 3 x + + 4< 16 and graph the solution on a number line. Step One: Rewrite the inequality in ax + b < c form. 3 x + + 4< x + < 1 3 x + 1 < 3 3 x + < 4 Step Two: Rewrite as two inequalities. x + < 4 AND x + > 4 Step Three: Solve each inequality. x < x > 6 Step Four: Write the solution as a compound inequality. 6< x < Step Five: Graph the solution on a number line. Ex: Solve the absolute value inequality 3 x < 5 and graph the solution on a number line. Step One: Rewrite as two inequalities. 3 x < 5 AND 3 x > 5 Step Two: Solve each inequality. x x 8 < < x > x < 8 FLIP! Step Three: Write the solution as a compound inequality. < x < 8 Step Four: Graph the solution on a number line. Page 14 of 15 McDougal Littell:

15 Writing an Absolute Value Inequality Ex: Write an absolute value inequality that has the solution shown o the number line below. Note: This compound inequality is an and statement with closed circles, so we will use an absolute value inequality of the form ax + b c. The endpoints are and 4. The half-way point between the two endpoints is 1. So, all of the possible values of the variable (x) must lie within 3 units of the half-way point. (i.e.: The difference between the values of x and 1 must be less than or equal to 3.) Solution: x 1 3 or 1 x 3 Application Problem with Absolute Value Inequalities Ex: At a bottling company, the machine accepts bottles only if the number of fluid ounces is 8 1 between 17 and 18 ounces. Let n be the number of fluid ounces that the machine accepts. Write an 9 9 absolute value inequality that describes all possible values of n. Note: The number of fluid ounces must be between two values, therefore we are looking for an and statement. So we will use an absolute value inequality of the form ax + b < c. The half-way point between and is 18. All of the possible values of n must lie within 1 9 of the half-way point. (i.e.: The difference between the values of n and 18 must be less than 1 9. Solution: 1 n 18 < or n < 9 On Your Own: Solve the absolute value inequality above and verify that is satisfies the problem. You Try: 1. Solve the absolute value inequality 5x and graph the solution on a number line.. Write an absolute value inequality that has the solution shown on the number line below. QOD: Explain how to determine if the solution to an absolute value inequality is an and or or compound inequality. Page 15 of 15 McDougal Littell:

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