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Solving Inequalities • To solve an inequality, use the same procedure as solving an equation with one exception. When multiplying or dividing by a negative number, reverse the direction of the inequality sign. • -3x < 6 divide both sides by -3 -3x/-3 > 6/-3 x > -2 Solutions…. You can have a range of answers…… -5 -4 -3 -2 -1 0 All real numbers less than 3 x< 3 1 2 3 4 5 Solutions continued… -5 -4 -3 -2 -1 0 All real numbers greater than -2 x > -2 1 2 3 4 5 Solutions continued…. -5 -4 -3 -2 -1 0 1 2 3 All real numbers less than or equal to 2 x2 4 5 Solutions continued… -5 -4 -3 -2 -1 0 1 2 3 4 5 All real numbers greater than or equal to -3 x 3 Did you notice, Some of the dots were solid and some were open? x2 -5 -4 -3 -2 -1 0 1 2 3 4 5 x0 -5 -4 -3 -2 -1 0 1 2 3 4 Why do you think that is? If the symbol is > or < then dot is open because it can not be equal. If the symbol is or then the dot is solid, because it can be that point too. Where is -1.5 on the number line? Is it greater or less than 2? -2 -25 -20 -15 -10 -5 0 5 • -1.5 is between -1 and -2. • -1 is to the right of -2. • So -1.5 is also to the right of -2. 10 15 20 25 Solve an Inequality w+5<8 We will use the same steps that we did with equations, if a number is added to the variable, we add the opposite sign to both sides: w + 5 + (-5) < 8 + (-5) w+0<3 w<3 All numbers less than 3 are solutions to this problem! THE TRAP….. When you multiply or divide each side of an inequality by a negative number, you must reverse the inequality symbol to maintain a true statement. Solving using Multiplication Multiply each side by the same positive number. 1 (2) x 3 (2) 2 x6 Solving Using Division Divide each side by the same positive number. 3x 9 3 3 x3 Solving by multiplication of a negative # Multiply each side by the same negative number and REVERSE the inequality symbol. (-1) x 4 (-1) Multiply by (-1). See the switch x 4 Solving by dividing by a negative # Divide each side by the same negative number and reverse the inequality symbol. 2x 6 -2 -2 x 3 Solving Inequalities • 3b - 2(b - 5) < 2(b + 4) 3b - 2b + 10 < 2b + 8 b + 10 < 2b + 8 -b + 10 < 8 -b < -2 b>2 0 1 2 More Examples x - 2 > -2 x + (-2) + (2) > -2 + (2) x+0>0 x>0 All numbers greater than 0 make this problem true! More Examples 4+y≤1 4 + y + (-4) ≤ 1 + (-4) y + 0 ≤ -3 y ≤ -3 All numbers from -3 down (including -3) make this problem true! Solving compound inequalities is easy if . . . . . . you remember that a compound inequality is just two inequalities put together. 5 2x 3 9 5 2x 3 2x 3 9 5 2x 3 9 You can solve them both at the same time: 5 2x 3 9 3 3 3 8 2x 6 2 2 2 4 x3 Write the inequality from the graph: -25 1: 2: 3: -20 -15 -10 -5 0 Write variable: boundaries: signs: 5 10 15 10 x 5 20 25 IsSolve this what the inequality: you did? 15 4 x 7 5 7 7 7 8 4 x 12 4 4 4 2 x 3 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 Solving Absolute Value Inequalities • Solving absolute value inequalities is a combination of solving absolute value equations and inequalities. • Rewrite the absolute value inequality. • For the first equation, all you have to do is drop the absolute value bars. • For the second equation, you have to negate the right side of the inequality and reverse the inequality sign. Solve: |2x + 4| > 12 2x + 4 > 12 2x > 8 x>4 or 2x + 4 < -12 2x < -16 x < -8 or x < -8 or x > 4 -8 0 4 Solve: 2|4 - x| < 10 |4 - x| < 5 4-x<5 -x<1 x > -1 and and 4 - x > -5 - x > -9 x<9 -1 < x < 9 -1 0 9