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CHAPTER 6 REVIEW Solving and Graphing Inequalities 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 “x < 5” means that whatever value x has, it must be less than 5. Try to name ten numbers that are less than 5! Numbers less than 5 are to the left of 5 on the number line. -25 -20 -15 -10 -5 0 5 10 15 20 25 • If 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. “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! Numbers greater than -2 are to the right of 5 on the number line. -25 -20 -15 -10 -5 0 5 10 15 20 25 -2 • If 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”. Where is -1.5 on the number line? Is it greater or less than -2? -25 -20 -15 -10 -5 0 5 10 15 20 25 -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. Solve an Inequality w+5<8 w + 5 + (-5) < 8 + (-5) w<3 All numbers less than 3 are solutions to this problem! More Examples 8 + r ≥ -2 8 + r + (-8) ≥ -2 + (-8) r ≥ -10 All numbers from -10 and up (including -10) make this problem true! More Examples x - 2 > -2 x + (-2) + (2) > -2 + (2) x>0 All numbers greater than 0 make this problem true! More Examples 4+y≤1 4 + y + (-4) ≤ 1 + (-4) y ≤ -3 All numbers from -3 down (including -3) make this problem true! There is one special case. ● Sometimes you may have to reverse the direction of the inequality sign!! ● That only happens when you multiply or divide both sides of the inequality by a negative number. Example: Solve: -3y + 5 >23 ●Subtract 5 from each side. -5 -5 -3y > 18 -3 -3 ●Divide each side by negative 3. y < -6 ●Reverse the inequality sign. ●Graph the solution. -6 0 Try these: 1.) Solve 2x + 3 > x + 5 -x -x 2.)Solve - c – 11 >23 + 11 x+3>5 -3 -3 -c > 34 -1 -1 x>2 c < -34 3.) Solve 3(r - 2) < 2r + 4 3r – 6 < 2r + 4 -2r -2r r–6<4 +6 +6 r < 10 + 11 You did to reverse . .remember .Good didn’tjob! you? the signs . . . 15 4 x 7 5 7 7 7 8 4 x 12 4 4 4 2 x 3 Example: 2x 6 4x 8 - 4x - 4x 2x 6 8 + 6 +6 2 x 14 -2 We turned the sign! -2 x 7 Ring the alarm! We divided by a negative! Remember Absolute Value Ex: Solve 6x-3 = 15 6x-3 = 15 or 6x-3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions! Ex: Solve 2x + 7 -3 = 8 Get the abs. value part by itself first! 2x+7 = 11 Now split into 2 parts. 2x+7 = 11 or 2x+7 = -11 2x = 4 or 2x = -18 x = 2 or x = -9 Check the solutions. Ex: Solve & graph. 4 x 9 21 • Becomes an “and” problem 15 3 x 2 -3 7 8 Solve & graph. 3x 2 3 11 • Get absolute value by itself first. 3x 2 8 • Becomes an “or” problem 3x 2 8 or 3x 2 8 3x 10 or 3x 6 10 x or x 2 3 -2 3 4 Example 1: This is an ‘or’ statement. (Greator). Rewrite. ● |2x + 1| > 7 ● 2x + 1 > 7 or 2x + 1 >7 ● 2x + 1 >7 or 2x + 1 <-7 ● x > 3 or In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. x < -4 Graph the solution. -4 3 Example 2: ● |x -5|< 3 This is an ‘and’ statement. (Less thand). ● x -5< 3 and x -5< 3 ● x -5< 3 and x -5> -3 ● ● Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. x < 8 and x > 2 2<x<8 Solve each inequality. Graph the solution. 2 8 Absolute Value Inequalities Case 1 Example: x 3 5 x 2 x 3 5 and x 3 5 x8 2 x 8 Absolute Value Inequalities Case 2 Example: 2 x 1 9 2x 1 9 2x 10 x 5 or 2x 1 9 2x 8 x4 x 5 OR x 4 Absolute Value • Answer is always positive • Therefore the following examples cannot happen. . . 3x 5 9 Solutions: No solution Graphing Linear Inequalities in Two Variables •SWBAT graph a linear inequality in two variables •SWBAT Model a real life situation with a linear inequality. Some Helpful Hints •If the sign is > or < the line is dashed •If the sign is or the line will be solid When dealing with just x and y. •If the sign > or the shading either goes up or to the right •If the sign is < or the shading either goes down or to the left When dealing with slanted lines •If it is > or then you shade above •If it is < or then you shade below the line Graphing an Inequality in Two Variables Graph x < 2 Step 1: Start by graphing the line x = 2 Now what points would give you less than 2? Since it has to be x < 2 we shade everything to the left of the line. Graphing a Linear Inequality Sketch a graph of y 3 Using What We Know Sketch a graph of x + y < 3 Step 1: Put into slope intercept form y <-x + 3 Step 2: Graph the line y = -x + 3