Linear Inequalities Math 10 Ms. Albarico Students are expected to: Express and interpret constraints using inequalities. Graph equations and inequalities and analyse graphs, both with and without graphing technology. Investigate and make and test conjectures about the solution to equations and inequalities using graphing technology. Solve problems using graphing technology. Write an inequality to describe its graph. Solve linear and simple radical, exponential, and absolute value equations and linear inequalities. Interpret solutions to equations based on context. Relate sets of numbers to solutions of inequalities. Key Terms Inequality Constraints Feasible Region System of Equation Optimal Solution Objective Function constraint a restriction on the allowable values of a variable in a problem feasible points data points that satisfy the constraints given feasible region a shaded region on a graph indicating that all points within the region are possible solutions to the problem feasible solution any solution to a problem that is possible within the constraints inequality a mathematical statement that shows that two numerical or variable expressions are not always equal graphical solutions solutions obtained by graphing the feasible region. intersection point the point where two graphs cross each other and where both graphs are equal objective function a function that allows you to find the maximum or minimum values using given constraints optimal solution the solution that best meets the constraints in the problem system of equations two or more equations involving the same variable quantities What s an Inequality? Is a range of values, rather than ONE set number An algebraic relation showing that a quantity is greater than or less than another quantity. Speed limit: Inequality Symbols Less than Greater than Less than or equal to Greater than or equal to Not equal to Solutions. You can have a range of answers All real numbers less than 2 x< 2 Solutions continued All real numbers greater than -2 x > -2 Solutions continued All real numbers less than or equal to 1 Solutions continued All real numbers greater than or equal to -3 Did you notice, Some of the dots were solid and some were open? 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. Write and Graph a Linear Inequality Sue ran a 2-K race in 8 minutes. Write an inequality to describe the average speeds of runners who were faster than Sue. Graph the inequality. Faster average speed > Distance Sue s Time Solving an Inequality Solving a linear inequality in one variable is much like solving a linear equation in one variable. Isolate the variable on one side using inverse operations. Add the same number to EACH side. x 3 < 5 Solve using addition: +3 x < 8 Solving Using Subtraction Subtract the same number from EACH side. -6 Using Subtraction Graph the solution Using Addition Graph the solution. 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. (2) Solving Using Division Divide each side by the same positive number. 33 Solving by multiplication of a negative # Multiply each side by the same negative number and REVERSE the inequality symbol. Multiply by (-1). (-1) See the switch Solving by dividing by a negative # Divide each side by the same negative number and reverse the inequality symbol. -2 Linear Inequality Inequality with one variable to the first power. for example: 2x-3 0 First, solve the inequality for x. 3 - x > 0 -x > -3 x < 3 Graph: x 0 3 1 > 0 2 > 0 True! The inequality has been graphed correctly More Examples system of equations two or more equations involving the same variable quantities Practice Exercise Class Activity Answer Investigation 1 & 2 on pages , Investigation 3, Part A & B on pages Investigation 4 on pages Homework CYU # 7, 8, and 9 on pages CYU # on pages CYU # on pages