ax + b < c ax + b c Graphing Inequalities:

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1 An inequality is a statement that contains one or more of the following symbols. < is less than is less than or equal to > is greater than is greater than or equal to is not equal to An inequality can be true, false, or neither true nor false. An inequality that contains a variable can be made true or false depending on the number that is substituted for the variable. A number that makes an inequality true is called a solution of the inequality, and we say that the number satisfies the inequality. A linear inequality in one variable can be written in one of the following forms where a, b, and c are real numbers and a 0. ax + b > c ax + b c ax + b < c ax + b c Checking if a number is a solution for an inequality: Is 9 a solution of 2x ? Is 2(9) true? Is true? Is true? No, 22 21, therefore 9 is NOT a solution A solution set of an inequality is the set of all numbers that make the inequality true. You can write solution sets in 2 ways: Graphing Inequalities: Two methods for graphing < or >: you can use an open circle or a parenthesis that opens in the direction where the inequality is true. Two methods for graphing or : you can use a closed circle or a bracket that opens in the direction where the inequality is true. If students get used to graphing with parentheses or brackets, it is a bit easier to teach them interval notation. Interval notation Interval notation is always written from left to right as it appears on a number line. If the interval you are graphing extends indefinitely to the left, that is designated. If the interval you are graphing extends indefinitely to the right, that is designated. In interval notation, and always have round brackets since, infinity is approached but never reached. In interval notation the smaller number appears first separated by a comma from the larger number.

2 Example:

3 Solving linear inequalities: Addition and Subtraction Properties of Inequality (this one is parallel to the property for equations): If a, b, c are real numbers, and a < b, then o a + c < b + c and o a c < b c. Examples: Solve and write answers in interval notation: (1) x 5 <7 (2) 3 + x > 4 (3) x (4) -3 < x + 2 5

4 Multiplication and Division Properties of Inequality: If a, b, c are real numbers, where c is positive (greater than 0), if a < b, then o a c < b c o a/c < b/c If a, b, c are real numbers, where c is negative (greater than 0), if a < b, then o a c > b c. o a/c > b/c Examples: (solve and write answers in interval notation) 1) 5x < 25 2) 8 >-2x 4) -x 5 5) -3 3x < 12 6) 2x 1 > 8 7) 4 x 8

5 Problem: A student has scores of 72%, 74%, and 78% on three exams. What percent score does he need on the last exam to earn a grade of no less than B (80%)?

6 Additional exercises to wrap your mind around it. Complete this table:

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8 Chapter Graphing Paired Data: Vocabulary: rectangular coordinate graph, rectangular coordinate system Cartesian coordinate system x-axis, y-axis, origin, coordinate plane, quadrants, ordered pair, x-coordinate, y-coordinate graphing, plotting interpolation, extrapolation Exercise: Example 1

9 Example 2: Gas mileage In 2007, the most fuel-efficient mid-sized car for city driving was the Toyota Prius, which was rated at 60 miles per gallon. The least efficient mid-sized car for city driving was the Ferrari 612 Scaglietti, rated at 10 miles per gallon. ( Prius Gallons (x) Mileage (y) Ferrari Gallons (x) Mileage (y)

10 3.2 Graphing linear equations: Vocabulary Linear equation in one variable, linear equation in two variables (when power is 1 and if variable is not multiplied or divided by another variable) Solution of an equation in two variables, table of solutions (or table of values) Graph of the equation, infinitely many solutions Standard form (general form) Exercises: 1) Creating a table of values using a linear equation:

11 2) Solving for y 3) Application Example: A group is going to hold a scrapbooking party as a fundraiser for a trip. They estimate that it will cost them $100 to create posters and flyers to advertise the event, and $300 to rent a large room for the event. They estimate that they will spend $10 on food for each participant. The organizer writes an equation for their costs: C = n and she can spend a total of $1,000 on the event. a) Assuming that the room is large enough, how many participants could be invited? b) If each participant pays $25 for the event, will they make any money? If so, how much?

12 4) 5) Fill in the missing values in the table of solutions for the linear equation, y = -x + 4, plot the points, and then draw the line through the points x y 3 5

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14 6) A student has created a table of solutions for each linear equation below: Plot the points that the student has found. Use the graphed points to identify which points (if any) are most likely incorrect. Circle the incorrect points in the table of values. Find the correct ordered pairs for any that are circled. Plot the corrected ordered pairs. (A) (B)

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