page 1 of 19 CLASS NOTES: 2 1 thru 2 3 and 1 1 Solving Inequalities and Graphing 1 1: Real Numbers and Their Graphs Graph each of the following sets. Positive Integers: { 1, 2, 3, 4, } Origin: { 0} Negative Integers: {, "4, "3, "2, "1} Integers: {, "4, "3, "2, "1, 0, 1, 2, 3, 4, } Real Numbers: The dot is called the graph. The number is called the coordinate. EX 1 Name the point that is the graph of the coordinate 5. EX 2 State the coordinate of the point A.
page 2 of 19 Inequality Symbols: > means is greater than < means is less than numbers to the LEFT < numbers to the RIGHT EX 3 Complete using > or < to make a true statement. (a) "5 " 4 (b) 1 2 1 3 (c) "3 " 4 (d) 7 0 (e) 8 " 9 (f) "5 1 4 2 1: Solving Inequalities in One Variable Definitions: Inequality - A statement that two expressions are NOT equal. Inequality Notation: < less than > greater than less than or equal to greater than or equal to not equal to
page 3 of 19 Hint: When graphing inequalities An open circle is used for the endpoint when graphing > or < or. EX 1 Graph each inequality. A closed circle is used for endpoint when graphing or. (a) x > 1 (b) x " 4 (c) x " 0 (d) "2 > x (e) " x < 3 (f) "5 # " x Hints: If you have the variable on the right, such as 6 < x. You can rewrite the inequality with the x on the left by CHANGING the DIRECTION of the inequality sign. x > 6 is the same as 6 < x In both cases, the inequality sign points to the 6 and large end opens to the x. If you have a negative variable such as " x > 7, you can rewrite the inequality with a positive variable by dividing both sides by "1 and CHANGING the DIRECTION of the inequality sign. x < "7 is the same as " x > 7
page 4 of 19 The Rules for Solving Inequalities are the same as the rules for solving equations EXCEPT for one thing NEW!! If you MULTIPLY or DIVIDE BOTH SIDES of the inequality by the same NEGATIVE number, REVERSE the DIRECTION of the inequality sign. EX 2 Solve each inequality and graph its solution set. (a) 5 x + 17 < 2 (b) 8 " 3 x # 20 Remember: CHANGE the DIRECTION of the inequality sign when you divide by "3.
page 5 of 19 EX 3 Solve each inequality and graph its solution set. (a) ( ) # 7 " x 5 3 " x (b) 2 x " 1 > x " 5 + 3 x (c) 24 " 2 x > 6 x REMEMBER this HINT: If you have the variable on the right, such as 6 < x. You can rewrite the inequality with the x on the left by CHANGING the DIRECTION of the inequality sign. x > 6 is the same as 6 < x In both cases, the inequality sign points to the 6 and large end opens to the x.
page 6 of 19 EX 4 Solve each inequality and graph its solution set. (a) " 1 2 x " 19 > "16 (b) 4 5 x " 4 > 4 " 4 5 x (c) 2 x 3 + 12 < x 6 + 18
page 7 of 19 NO SOLUTION or ALL REAL NUMBERS Question: Answer: What does it mean if I am solving an inequality, and all of my variables go away? The inequality has either (a) (b) NO solution or ALL REAL NUMBERS as its solution EX 5 Solve each inequality. A no solution example: An all real numbers example: 2 x < " 1 ( 2 "6 " 4 x ) (a) 4 x > 2 ( 3 + 2 x ) 4 x > 6 + 4 x "4 x = " 4 x 0 > 6 (b) 2 x < 6 2 + 4 x 2 2 x < 3 + 2 x "2 x = " 2 x 0 < 3 Wait! No more x s! Wait! No more x s! And 0 is NOT greater than 6 And 0 IS less than 3 This inequality is NOT TRUE!! This inequality is TRUE!! ** NO SOLUTION ** **ALL REAL NUMBERS are solutions** EX 6 State whether each inequality is TRUE or FALSE. (a) 6 > "7 (b) 0 < "3 (c) 0 " 0
page 8 of 19 EX 7 Solve each inequality. (a) x " ( 7 " x ) > 2 x + 1 (b) 3( 4 x " 1) > 2 ( 6 x " 5) (c) "( 10 " x ) < x " 15 (d) 4 x " 2 ( 3 + 2 x )
page 9 of 19 2 2: Solving Combined Inequalities Definitions: Combined Inequality - Two or more inequalities connected with the word(s) AND or OR. There are TWO types of combined inequalities. Conjunction - Disjunction - Two or more inequalities combined with the word AND. Two or more inequalities combined with the word OR. Comparison of COMBINED INEQUALITIES CONJUNCTION and DISJUNCTION or How is a conjunction or disjunction written? x > "1 and x < 3 x > "1 or x < 3 Note: Conjunctions can also be written as triples using LESS THAN symbols. Note: Disjunctions cannot be written any other way. EX: x > "1 means the same as "1 < x So, our example can also be written as: "1 < x and x < 3 "1 < x < 3 Triples must always be written with LESS THAN symbols, and the lesser number must always be on the LEFT. EX: Rewrite each conjunction as a triple. (a) x < 4 and x " 1 (b) x " 3 and x < 7
page 10 of 19 CONJUNCTION and DISJUNCTION or How do we solve conjunctions and disjunctions? Find the value(s) of the variable for which both inequalities are true. Find the value(s) of the variable for which at least one of the inequalities are true. How do we solve conjunctions and disjunctions GRAPHICALLY? x > "2 and x < 4 x > "2 or x < 4 1 st : Graph each of the two inequalities separately. 1 st : Graph each of the two inequalities separately. 2 nd : Graph the set of points where the two separate graphs overlap. This is the graph of your combined solution. 2 nd : Graph the set of all points that are part of at least one of the separate graphs. This is the graph of your combined solution. 3 rd : Write an inequality for the graph of your combined solution. 3 rd : Write an inequality for the graph of your combined solution. EXAMPLES to GRAPH EX 1 "1 # x < 3 EX 2 x " #1 or x < 3 Solution: Solution:
page 11 of 19 CONJUNCTION and EX 3 x " #1 and x > 3 DISJUNCTION or EX 4 x " #1 or x > 3 Solution: Solution: EX 5 x " #1 and x > 3 EX 6 x " #1 or x > 3 Solution: Solution: EX 7 x " #1 and x < 3 EX 8 x " #1 or x < 3 Solution: Solution:
page 12 of 19 Question: Answer: How do we solve combined inequalities that appear too complex to solve graphically? Solve each of the inequalities separately. Then use the two solutions to solve the combined inequality graphically. Remember There are TWO types of COMBINED INEQUALITIES conjunctions and disjunctions. CONJUNCTIONS: Inequalities connected with the word AND. EX 9 Solve each combined inequality. Examples: 2 < x " 7 3 < 2 x + 5 " 15 ( ) > 4 and 2 x + 3 < 5 " x " 1 (a) "7 # 5 x " 2 < 8 1 st : If the AND inequality is a TRIPLE, you may be able to solve it by applying the Golden Rule of Algebra to ALL THREE PARTS. For example, first add 2 to all three parts of the inequality. "7 # 5 x " 2 < 8 2 nd : Graph the combined solution.
page 13 of 19 (b) "2 # " x + 3 < 4 1 st : Apply the Golden Rule of Algebra to ALL THREE PARTS. "2 # " x + 3 < 4 Caution!! (1) x must be positive in your final solution, so you will need to divide all three parts by "1. (2) Change the direction of BOTH inequality signs when you divide all three parts by "1. (3) Your inequality signs must be LESS THANS in a triple. If you have >, flip your entire inequality 5 " x > #1 2 nd : Graph the combined solution. #1 < x $ 5 (c) "7 < 3 4 x " 7 < "4 1 st : Apply the Golden Rule of Algebra to ALL THREE PARTS. "7 < 3 4 x " 7 < "4 2 nd : Graph the combined solution.
page 14 of 19 (d) "( x " 1) > 4 and " ( 1 " x ) < 4 1 st : Solve each of the two inequalities separately. Be sure to get x by itself on the left side for each. 2 nd : Graph each of the two separate solutions in order to solve the combined inequality graphically. 3 rd : The solution is where the two graphs overlap. Graph the combined solution. 4 th : Write an inequality for the graph of your combined solution. (e) Why might this conjunction have no solution?? "2 > x " 2 > 2
page 15 of 19 DISJUNCTIONS: Inequalities connected with the word OR Examples: x > 2 or x " 0 x + 7 < 4 or 7 " x < 1 EX 10 Solve each combined inequality. (a) x + 7 < 4 or 7 " x < 1 1 st : Solve each of the two inequalities separately. Be sure to get x by itself on the left side for each. 2 nd : Graph each of the two separate solutions in order to solve the combined inequality graphically. 3 rd : The solution is all points that are part of at least one of the two graphs. Graph the combined solution. 4 th : Write an inequality for the graph of your combined solution.
page 16 of 19 (b) 7 " 2 x # 1 or 3 x + 10 < 4 " x 1 st : Solve separately. 2 nd : Graph each separately. 3 rd : Graph the combined solution. (What points are part of at least one graph?) 4 th : Write an inequality for the combined graph.
page 17 of 19 (c) 1 2 x + 2 > 5 or " 1 3 x < 2 Your final solution: (d) 2 x + 1 < "9 or 0.1 x + 0.5 # 0 Your final solution:
page 18 of 19 2 3: Problem Solving Using Inequalities Greater Than Phrase A number is three. The number is ten. Translation Less Than The number is twelve. The number is five. Conjunction The number is twenty and thirty. The number is five and ten, inclusive. Sample Problems: A. A store sells calculators for $8 each. It costs the store $5.50 to purchase each calculator plus a delivery charge of $25 for each order. How many calculators must be ordered and sold to produce a profit of at least $80? Question: Answer: Formula: (Sales) (Cost of Sales) = (Profit) unit $ " quantity + delivery = total $ Sales Cost of Sales TOTAL
page 19 of 19 B. Find all sets of 4 consecutive integers whose sum is between 10 and 20. Question: Answer: Integers 1 st integer 2 nd integer 3 rd integer 4 th integer SUM C. A regular hexagon, a square, and an equilateral triangle all have equal sides. If the sum of the perimeters of the square and the triangle is no more than 18cm less than twice the perimeter of the hexagon, what is the minimum length of each side? Remember: Perimeter is the sum of the lengths of the sides. Question: Answer: Hint Formula: (P S + P T! " P H # ) length of sides " # of sides = PERIMETER Square Triangle Hexagon