Algebra I Solving & Graphing Inequalities 2016-01-11 - PDF document

Slide 1 / 182 Slide 2 / 182 Algebra I Solving & Graphing Inequalities 2016-01-11 www.njctl.org Slide 3 / 182 Table of Contents click on the topic to go Simple Inequalities Addition/Subtraction to that section Simple Inequalities Multiplication/Division Two-Step and Multiple-Step Inequalities Solving Compound Inequalities Special Cases of Compound Inequalities Graphing Linear Inequalities in Slope-Intercept Form Solving Systems of Inequalitites Glossary & Standards

Slide 4 / 182 Simple Inequalities Involving Addition and Subtraction Return to Table of Contents Slide 5 / 182 Inequality An Inequality is a mathematical sentence that uses symbols, such as <, ≤, > or ≥ to compare to quantities. Slide 6 / 182 What do these symbols mean? (when read from LEFT to RIGHT) Less Than Less or Equal To Than click Greater Greater Than Than or Equal To click

Slide 7 / 182 Slide 8 / 182 Inequality Write an inequality for the sentence below: Three times a number, n, is less than 210. Click The sum of a number, n, and fifteen is greater than or equal to nine. Click Slide 9 / 182 Graphing Inequalities Remember! Open circle means that number is not included in the solution set and is used to represent < or >. Closed circle means the solution set includes that number and is used to represent ≤ or ≥.

Slide 10 / 182 Solving Inequalities · Solving one-step inequalities is much like solving one-step equations. · To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations. Slide 11 / 182 Isolate the Variable To find the solution, isolate the variable x. Remember, it is isolated when it appears by itself on one side of the equation. Slide 12 / 182

Slide 13 / 182 Solving Inequalities Step 2: Decide whether or not the circle on your boundary should be open or closed based on the symbol used. You can check the computation by substituting the end point of 6 for x. In this case, the end point is not included (open circle) since x < 6. -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 Slide 14 / 182 Slide 15 / 182 Review of Solving Inequalities Using Addition and Subtraction The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at: http://www.njctl.org/courses/math/7th-grade/ equations-inequalities-7th-grade/

Slide 16 / 182 1 Which graph is the solution to the inequality: a number, n, minus is greater than one third? 5 2 6 A 1 -5 -4 -3 -2 -1 0 2 3 4 5 5 2 6 B 1 -5 -4 -3 -2 -1 0 2 3 4 5 5 2 6 C 1 -5 -4 -3 -2 -1 0 2 3 4 5 2 5 6 D -5 -4 -3 -2 -1 0 1 2 3 4 5 Slide 17 / 182 2 Which graph is the solution to the inequality ? A -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 B -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 D -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 18 / 182 3 Which graph is the solution to the inequality ? A -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 B -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 D -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Slide 19 / 182 4 Which graph is the solution to the inequality ? A -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 B -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 D -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 20 / 182 5 Which graph is the solution to the inequality ? 1.5 A -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1.5 B -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1.5 C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 1.5 D -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 21 / 182 Simple Inequalities Involving Multiplication and Division Return to Table of Contents

Slide 22 / 182 Inequalities Involving Multiplication and Division Again, similarly to solving equations, we can use the properties of multiplication and division to solve and graph inequalities - with one minor difference, which we will encounter in the upcoming slides. Slide 23 / 182 Multiplying or Dividing by a Positive Number Since x is multiplied by 3, divide both sides by 3 to isolate the variable. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 24 / 182

Slide 25 / 182 Review of Solving Inequalities Using Multiplication and Division The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at: http://www.njctl.org/courses/math/7th-grade/ equations-inequalities-7th-grade/ Slide 26 / 182 6 Which graph is the solution to the inequality, the product of 4 and a number, x, is greater than 24? A -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 B -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 D -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 27 / 182

Slide 28 / 182 Slide 29 / 182 9 Find the solution to the inequality. A B C D Slide 30 / 182 10 Find the solution to the inequality. A B C D

Slide 31 / 182 Multiplying or Dividing by a Negative Number So far, all the operations we have used worked the same as solving equations. The difference between solving equations versus inequalities is revealed when multiplying or dividing by a negative number. The direction of the inequality changes only if the number you are using to multiply or divide by is negative . Slide 32 / 182 Solve and Graph *Note: Dividing each side by -3 changes the ≥ to ≤. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 click for answer Slide 33 / 182 Solve the inequality and graph the solution. 11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Slide 34 / 182 12 Solve the inequality and graph the solution. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 35 / 182 13 Solve the inequality and graph the solution. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 36 / 182 14 Solve the inequality and graph the solution. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Slide 37 / 182 Summary In review, an inequality symbol stays the same direction when you: · Add, subtract, multiply or divide by the same positive number on both sides. · Add or subtract the same negative number on both sides. An inequality symbol changes direction when you: · Multiply or divide by the same negative number on both sides. Slide 38 / 182 Solving Two-Step and Multiple-Step Inequalities Return to Table of Contents Slide 39 / 182 Inequalities Now we'll solve more complicated inequalities that have multi-step solutions because they involve more than one operation. Solving inequalities is like solving a puzzle. Keep working through the steps until you get the variable you're looking for alone on one side of the inequality using the same strategies as solving an equation.

Slide 40 / 182 Slide 41 / 182 Multiplying or Dividing by a Negative Number Another reminder! If you multiply or divide by a negative number, reverse the direction of the inequality symbol! Slide 42 / 182

Slide 43 / 182 Two Step Inequalities Example: Solve the inequality and graph the solution. Add 9 to both sides Divide both sides by 4 (sign stays the same) -1 1 2 3 4 5 6 -10 -10 -9 -9 -8 -8 -7 -7 -6 -6 -5 -5 -4 -4 -3 -3 -2 -2 -1 0 0 1 2 3 4 5 6 7 7 8 9 10 8 9 10 click for answer Slide 44 / 182 Solve and Graph Try these. Solve each inequality and graph each solution. 1. -1 1 2 3 4 -10 -9 -8 -7 -6 -5 -4 -3 -2 0 5 6 7 8 9 10 2. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 45 / 182 Solve and Graph Try these. Solve each inequality and graph the solution. 3. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 4. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

Slide 46 / 182 15 Solve and graph the solution. A 2.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 B 2.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 C 2.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 D 2.5 -5 -4 -3 -2 -1 0 1 2 3 4 5 Slide 47 / 182 Slide 48 / 182

Slide 49 / 182 Slide 50 / 182 19 Solve and graph the solution. A -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 B -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 D -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 51 / 182 20 Which graph represents the solution set for: Question from ADP Algebra I End-of-Course Practice Test A -2 -1 2 0 1 B -2 -1 0 1 2 C -2 -1 2 0 1 D -2 2 -1 0 1