Presentation on theme: "Solving and Graphing Inequalities"— Presentation transcript:
1 Solving and Graphing Inequalities CHAPTER 6 REVIEWSolving and Graphing Inequalities
2 ≥ : greater than or equal to An inequality is like an equation, but instead of an equal sign (=) it has one of these signs:< : less than≤ : less than or equal to> : greater than≥ : greater than or equal to
3 “x < 5” means that whatever value x has, it must be less than 5. Try to name ten numbers that are less than 5!
4 Numbers less than 5 are to the left of 5 on the number line. 51015-20-15-10-5-252025If you said 4, 3, 2, 1, 0, -1, -2, -3, etc., you are right.There are also numbers in between the integers, like 2.5, 1/2, -7.9, etc.The number 5 would not be a correct answer, though, because 5 is not less than 5.
5 Try to name ten numbers that are greater than or equal to -2! “x ≥ -2”means that whatever value x has, it must be greater than or equal to -2.Try to name ten numbers that are greater than or equal to -2!
6 Numbers greater than -2 are to the right of 5 on the number line. 51015-20-15-10-5-252025-2If you said -1, 0, 1, 2, 3, 4, 5, etc., you are right.There are also numbers in between the integers, like -1/2, 0.2, 3.1, 5.5, etc.The number -2 would also be a correct answer, because of the phrase, “or equal to”.
7 Where is -1.5 on the number line? Is it greater or less than -2? 51015-20-15-10-5-252025-2-1.5 is between -1 and -2.-1 is to the right of -2.So -1.5 is also to the right of -2.
8 All numbers less than 3 are solutions to this problem! Solve an Inequalityw + 5 < 8w (-5) < 8 + (-5)All numbers less than 3 are solutions to this problem!w < 3
9 All numbers from -10 and up (including -10) make this problem true! More Examples8 + r ≥ -28 + r + (-8) ≥ -2 + (-8)r ≥ -10All numbers from -10 and up (including -10) make this problem true!
10 All numbers greater than 0 make this problem true! More Examplesx - 2 > -2x + (-2) + (2) > -2 + (2)x > 0All numbers greater than 0 make this problem true!
11 All numbers from -3 down (including -3) make this problem true! More Examples4 + y ≤ 14 + y + (-4) ≤ 1 + (-4)y ≤ -3All numbers from -3 down (including -3) make this problem true!
12 There is one special case. Sometimes you may have to reverse the direction of the inequality sign!!That only happens when youmultiply or divide both sides of the inequality by a negative number.
13 Example: -3y > 18 Solve: -3y + 5 >23 -5 -5 -3y > 18y < -6Subtract 5 from each side.Divide each side by negative 3.Reverse the inequality sign.Graph the solution.-6
18 * Plug in answers to check your solutions! Ex: Solve 6x-3 = 156x-3 = or 6x-3 = -156x = or 6x = -12x = 3 or x = -2* Plug in answers to check your solutions!
19 Get the abs. value part by itself first! Ex: Solve 2x = 8Get the abs. value part by itself first!2x+7 = 11Now split into 2 parts.2x+7 = 11 or 2x+7 = -112x = 4 or 2x = -18x = 2 or x = -9Check the solutions.
21 Solve & graph. Get absolute value by itself first. Becomes an “or” problem
22 Example 1: |2x + 1| > 7 2x + 1 > 7 or 2x + 1 >7 This is an ‘or’ statement. (Greator). Rewrite.In the 2nd inequality, reverse the inequality sign and negate the right side value.Solve each inequality.Graph the solution.|2x + 1| > 72x + 1 > 7 or 2x + 1 >72x + 1 >7 or 2x + 1 <-7x > 3 or x < -43-4
23 Example 2: This is an ‘and’ statement. (Less thand). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value.Solve each inequality.Graph the solution.|x -5|< 3x -5< 3 and x -5< 3x -5< 3 and x -5> -3x < 8 and x > 22 < x < 882
25 Absolute Value Inequalities Case 2 Example:orOR
26 Solutions: No solution Absolute ValueAnswer is always positiveTherefore the following examplescannot happen. . .Solutions: No solution
27 Graphing Linear Inequalities in Two Variables SWBAT graph a linear inequality in two variablesSWBAT Model a real life situation with a linear inequality.
28 Some Helpful Hints If the sign is > or < the line is dashed If the sign is or the line will be solidWhen dealing with just x and y.If the sign > or the shading either goes up or to the rightIf the sign is < or the shading either goes down or to the left
29 When dealing with slanted lines If it is > or then you shade aboveIf it is < or then you shade below the line
30 Graphing an Inequality in Two Variables Graph x < 2Step 1: Start by graphing the line x = 2Now what points would give you less than 2?Since it has to be x < 2 we shade everything to the left of the line.
31 Graphing a Linear Inequality Sketch a graph of y 3
32 Step 1: Put into slope intercept form Using What We KnowSketch a graph of x + y < 3Step 1: Put into slope intercept formy <-x + 3Step 2: Graph the line y = -x + 3