求解：并集（OR）

(Video) Writing OR Compound Inequalities Using Interval Notation

by PatrickJMT

This video by PatrickJMT covers "or" compound inequalities.

$\large{\text{Summary}}$

An "or" inequality requires a solution that satisfies any of the individual parts. Any number that solves one part of an "or" inequality satisfies the whole thing. Let's look at a compound inequality that we don't have to solve.

Any number that is less than $8$ or greater than or equal to $20$ satisfies the inequality. Anything from $[8,20)$ won't satisfy the inequality.

If you want to review interval notation for inequalities, check out this lesson.

Solving Linear Compound Inequalities

Compound inequalities involve two separate parts. Some inequality statements require you to find solutions that work in both parts. These are AND inequalities. Some inequalities require solutions that work in one OR both of the parts. It's kind of like finding the union of sets! Let's look at one of those.

$$6-3x\le 21 \text{ OR } x+2\le 6$$

We can use the skills we learned when producing equivalent inequalities in order to solve each part of the inequality on its own, using subtraction and division! $$\begin{align} 6-3x&\le 21 \color{#8925AE}{\text{ Subtract 6 from both sides }}\\ -3x&\le 15 \color{#8925AE}{\text{ Divide both sides by -3 and flip the sign }}\\ x&\ge -5\\ \\ x+2&\le 6 \color{#8925AE}{\text{ Subtract 2 from both sides }}\\ x&\le 4\\ \\ \text{So, }x\ge -5 &\text{ OR } x\le 4 \end{align}$$

We can put this inequality on a number line.

After graphing both inequalities on this number line, we can see that the entire number line is highlighted! That means that any real number can be a solution to this compound inequality. Let's dive a little deeper on how to analyze our compound inequalities using interval notation.

Remember interval notation? We can use it to help analyze our OR inequality! Notice that the area from $[-5,4]$ is shared by both inequalities. In some cases this area would be our entire answer, but "or" inequalities are a special case. Any number only needs to satisfy one of the two final inequalities. The numbers in the interval $[-5,4]$ satisfy both inequalities, but the numbers outside of that interval satisfy at least one. This means that the solution to the compound inequality is all real numbers. (Here is a refresher on real numbers if you need it!)

Not all OR compound inequalities are going to work like that, so still be mindful when you're solving future problems!

Solving OR Inequalities

Earlier, we saw how to solve AND inequalities, with the solution being the intersection of two inequalities. Now, we'll look at OR inequalities.

To solve a compound OR inequality, first solve each component of the OR inequality separately. Then, you can graph your solution if you want to visualize the solutions. Remember: any value that satisfies either of the inequalities is a valid solution!

As we see in the above graphic, we first want both inequalities to be in standard form. With a little math, we can make the inequalities:

$$x≤-3$$ $$x≥1$$

Then, we graph both inequalities as normal, ensuring that both circles are filled (because it's greater than or equal to and less than or equal to).

Once we graph the solutions, we know that any number satisfying either inequality is a valid solution.