Section 1 B: Compound Inequalities Some inequality statements contain inequalities with an and or an or between them. We call these type of inequalities compound inequalities. x > and x < x < and 8 x x or > x x or x Graphing the solution to a single Inequality Graph : x > When we graph the solution to a single inequality like x > we shade in all the points that make this inequality true. These are all the points greater than Graph : x When we graph the solution to a single inequality like x we shade in all the points that make this inequality true. These are all the points less than or equal to x > x Graphing the solution to a compound Inequality with an AND Graph the solution to x > AND x < The solution to x > AND x < are all the points that will make both x > AND x < true at the same time. To find these points we must look for the points that are on BOTH the x > graph AND the x < graph. If the graphs are graphed on the same line then where the graphs is the solution. It is hard to see the with both graphs on top of each other so we graph each separate graph one above the other and look for the area where the graphs. x > x < the points where the graphs are between and but the and the are NOT included. The solution graph is the set of points between and but the endpoints are NOT included. Answer: < x < Interval Notation: x (, ) Math 10 Section 1 B Page 1 011 Eitel
The AND Case 1. Graph each inequality on a separate number line one under the other.. Observe where the graphs. Think of moving the graphs onto the same line and think where the shaded areas would touch. The area of is the solution. Example 1 Example x > and 7 > x x and 8 x x > 7 7 > x x 8 8 x the points where the graphs are The points where the graphs are between and 7 between and 8 but the and the 7 are NOT included and the and the 8 ARE included < x < 7 x 8 7 8 Interval Notation: x (, 7 ) Interval Notation: x [,8 ] Example Example x and x < 9 x > and 1 x x 9 x < 9 x > 1 1 x the points where the graphs are The points where the graphs are between and 9 between and 1 the IS included but the 9 IS NOT included the IS NOT included but the 1 IS included x < 9 < x 1 9 1 Interval Notation: x [, 9 ) Interval Notation: x (, 1 ] Math 10 Section 1 B Page 011 Eitel
Example Example x and x <1 x > 9 and x x x > 9 9 1 x <1 x the points where the graphs are The points where the graphs are to the left of to the right of 9 and the IS included but the 9 is NOT included x x > 9 9 Interval Notation: x (, ] Interval Notation: x ( 9, + ) Example 7 x and x 7 x there is no 7 x 7 there are NO POINTS where the graphs so there is No Solution Math 10 Section 1 B Page 011 Eitel
Graphing the Solution to Compound Inequality with an OR Graph the solution to x < or x The solution to x < OR x are all the points that will make x < true combined with all the points that make x true. The OR case requires we take ALL THE POINTS on either of the separate graphs instead of the like we did with the and case. If the graphs were graphed on the same line we would take any point that was shaded on any part of the graph. We graph each separate graph one above the other and look for any points on either graph. take all the points on either graph x < x we take the points to the left ot combined with the points to the right ot and the IS included. Answers: x < or x Interval Notation: x (, ) U [, + ) Math 10 Section 1 B Page 011 Eitel
The OR Case 1. Graph each inequality on a separate number line one under the other.. Observe all the points where there are points that are shaded. Think of moving the graphs onto the same line and observe where the shaded areas. Any shaded area is part of the solution. All the points on each of the separate graphs are part of the solution. Example 1 Example x < or x > x or x 8 8 The points to the left of are shaded The points to the right of are shaded All the points on each of the separate graphs are part of the solution The points to the left of including are shaded The points to the right of 8 including 8 are shaded All the points on each of the separate graphs are part of the solution x (, ) U (, + ) x (, ] U [ 8, + ) Example Example x > or x x or x > x > x x x > the point and all the points to the right of the points to the right of x > or x x > or x x [, + ) x (, + ) Math 10 Section 1 B Page 011 Eitel
Example Example x < or x x or x < x < x < x x the point and all the points to the left of the points to the left of x < or x x or x < x (, ] x (, ) Example 7 Example 8 x or x < x > or x < x < x > x x < All Points are shaded by at least one of the graphs All Points are shaded by at least one of the graphs All Real Numbers All Real Numbers x (, + ) x (, + ) Math 10 Section 1 B Page 011 Eitel