Solving Compound Inequalities Using AND or OR
Description/Explanation/Highlights
Video Description
This video explains how to solve compound inequalities using <i>AND</i> or <i>OR</i>.
Steps and Key Points to Remember
To solve a compound inequality involving <i>AND</i> or <i>OR</i>, follow these steps:
- A problem written in the format \(2\leq 2x-2<8\) is really an AND problem and means the same thing as \(2x-2\geq 2\) and \(2x-2<8\)
- The above problem is generally read as “2x minus 2 is between 2 and 8, including 2.”
- To solve a compound AND problem, look at the middle expression and isolate x as you would to solve any equation or inequality making sure to do the same thing to the middle and both the left and right sides of the inequality.
- In this example, add 2 to the middle to get rid of -2 and to 2 on the left as well as 8 on the right. The inequality will now read: \(4\leq 2x<10\).
- Next, get rid of the 2 in front of x by dividing it by 2 and also dividing 4 & 10 by 2. The result will be \(2\leq x <5\) and is read “x is between 2 and 5, including 2.”
- Although it doesn’t occur in this example, be sure to flip the inequality signs if you multiply or divide by a negative number as with all inequalities.
- Compound inequalities with OR are easily recognized since they have the OR included in them. For example: \(-3x-1\leq 8 \text{ or }-2x-5\geq 9\)
- To solve compound inequalities with OR, solve them as separate problems with the word OR between the answers. In the above example: \(x\geq-3\text{ or }x\leq -7\).
- Notice in the above example that both inequality signs were flipped since solving both involved dividing by a negative number.
Here are some key points to keep in mind when solving compound inequalities:
- AND represents between and usually is written without the word AND but with an expression between two inequality signs.
- When solving an AND inequality, do the Algebra to all three parts: the middle, the left, and the right.
- Remember if you multiply or divide by a negative number, as in all inequalities, flip the inequality sign or signs.
- OR inequalities are solved as two separate inequalities with the answers connected with the word OR.
- Sometimes it helps to draw and shade the results on a number line to visualize.
Video Highlights
- 00:00 Introduction
00:22 \(2\leq 2x-2 <8\) example of AND inequality
02:50 \(11<-4x+3<19 \) second example of AND inequality
04:25 \(-3x-1\leq 8\text{ OR }-2x-5\geq 9\) example of OR inequality
08:20 Conclusion
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