Objectives: Review open, closed, and mixed intervals, and begin discussion of graphing points in the xyplane. Interval notation

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1 MA 0090 Section 18 - Interval Notation and Graphing Points Objectives: Review open, closed, and mixed intervals, and begin discussion of graphing points in the xyplane. Interval notation Last time, we looked at graphing linear inequalities. The inequality (1) x 2 has a graph that looks like the following. There are other kinds of graphs that are commonly used. We might have a three-sided (or double) inequality like (2) 2 < x 3 Here, the solution set are all the x s that are between 2 and 3 including 3. The graph in this case looks like the following. Note that at the 2, we have a strict less than, so we don t want the 2, and we have a hollow dot at 2. There is a less than or equal to at the 3, so we have a solid dot at the 3. Both of the sets described so far are examples of intervals. In general, intervals are sets of real numbers that form a continuous set in one piece. A common notation for intervals describes the inequality 2 < x 3 as (3) ( 2, 3] Again, this means that we want all the numbers between 2 and 3, and the rounded bracket at the 2 means that we don t want the 2, and the square bracket at the 3 means that we want the 3. In other words, the rounded bracket goes with hollow dots, and the square brackets go with solid dots. For the inequality x 2, we use the interval notation [ 2, ). These are all the x s greater than or equal to 2. These are also less than positive infinity. Since is not really a number, it won t be included in any interval, and therefore, won t get a solid dot nor a square bracket. In other words, we ll always use a rounded bracket with infinities. For graphs that go to infinity to the left, we ll say that these go towards negative infinity and use the symbol. The inequality (4) x < 2 will be written as (5) (, 2) in interval notation. The corresponding graph looks like the following. 1

2 MA 0090 Section 18 - Interval Notation and Graphing Points 2 Quiz 18, Part I Write the following inequalities in interval notation x 5 [1, 5] 2. 4 < x 2 ( 4, 2] 3. x 3 (, 3] 4. x > 4 (4, ) Graph the sets described in interval notation. 5. [3, 5) solid on 3, hollow on 5, and bar in between. 6. (0, 3) hollow on 0, hollow on 3, and bar in between. 7. [0, 1] solid on 0, solid on 1, and bar in between. 8. [ 2, ) solid on 2, and arrow going to right. 9. (, 0) hollow on 0, and arrow going to left. Graphing points in the plane We ll be working with equations in two variables, and with those, we ll often want to draw graphs describing certain relationships. We may want to talk about a situation where we have two variables x and y, and x = 2 and y = 3. We ll use the shorthand notation (x, y) = (2, 3), or simply (2, 3). The notation (2, 3) looks like interval notation, but is very different. In this case, we have an ordered pair or the coordinates for a point in the xy-plane. We graph a point using two numbers lines, one for the x and one for the y. For the point, (2, 3), we have x = 2 and y = 3. We look for the 2 on the x-axis, and the 3 on the y-axis. We ll put a dot above the x = 2 and at the same height as the y = 2.

3 MA 0090 Section 18 - Interval Notation and Graphing Points 3 If we wanted to graph the point with coordinates ( 2, 1), we would look above x = 2 and at the same height as y = 1. Quiz 18, Part II 10. What are the coordinates of the point shown in the following graph? (2, 2) 11. What are the coordinates of the point shown in the following graph? ( 2, 3)

4 MA 0090 Section 18 - Interval Notation and Graphing Points Graph the point with coordinates (1, 3). Above 1 on the x-axis, and to the right of 3 on the y-axis. 13. Graph the point with coordinates ( 3, 2). Above 3 on the x-axis, and to the left of 2 on the y-axis. 14. Graph the point with coordinates (0, 0). The point lies at the place where the two axes cross. This is called the origin. 15. Graph the point with coordinates (1, 1). Below the 1 on the x-axis, and to the right of the 1 on the y-axis. Homework 18 Write the following inequalities in interval notation x < x 4 3. x 3 4. x < 4 Describe the set shown in interval notation (continued on next page)

5 MA 0090 Section 18 - Interval Notation and Graphing Points What are the coordinates of the point shown in the following graph? 11. What are the coordinates of the point shown in the following graph? 12. What are the coordinates of the point shown in the following graph?