Sign up for your FREE 7-day trial.Get instant access to all the benefits of SparkNotes PLUS! Cancel within the first 7 days and you won't be charged. We'll even send you a reminder.

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Step 2 of 4

Choose Your Plan

Step 3 of 4

Add Your Payment Details

Step 4 of 4

Payment Summary

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thanks for creating a SparkNotes account! Continue to start your free trial.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

When we see a statement like "x < 7 or x≥11", written in set notation as {x : x < 7orx≥11}, the word or denotes the union of the two sets of numbers which satisfy each inequality. Thus, {x : x < 7orx≥11} = {x : x < 7}∪{x : x≥11}. This is the set of values which satisfy eitherx < 7orx≥11. The value 5 satisfies the statement, as does the value 14.

We can graph the union of two inequalities on the number line. To do this, simply graph both inequalities:

Every point on the dark line is a member of the set {x : x < 7orx≥11}.

Sometimes the two inequalities will overlap. This is fine. The set of all values which satisfy either inequality is the set of all points which satisfy one or the other or both--this includes the overlap.

Intersection of Inequalities

When we see a statement like "0≤x < 4", also written as "0≤x and x < 4", or as {x : 0≤x < 4}, the compound inequality or the word and denotes the intersection of the two sets of numbers which satisfy each inequality. Thus, {x : 0≤x < 4} = {x : 0≤x}∩{x : x < 4}. This is the set of values which satisfy both0≤xandx < 4. The value 2 satisfies the statement, but the value -3 does not, and the value 5 does not.

We can graph the intersection of two inequalities on the number line. To do this, lightly graph each inequality. Then darken the line which appears in the graph of both inequalities. Finally, erase the light line which does not appear in the graph of both inequalities:

Every point on the dark line is a member of the set {x : 0≤x < 4}.