不等式：绝对值表达式

(Video) Matching Number Lines with Absolute Value Inequalities

by PatrickJMT

This video by Patrick JMT Hows how to turn absolute value inequalities into compound inequalities.

$\large{\text{Summary}}$

He starts out by graphing the absolute value inequality: $|x|>9$.

The absolute value symbol makes negative numbers positive.

Let's start by looking at what positive numbers (and zero) satisfy the inequality. We can see that $0$ through $8$ won't work since they're all less than $9$. And, $9$ won't work either since the inequality is strictly greater than. But $10$ would work since it's greater than $9$. So we can graph:

Let's look at the negative numbers: $-1$ through $-8$ would simplify to $1$ through $8$ respectively. We know those won't work. We also know we are looking for numbers strictly greater than $9$. The absolute value of $-9$ is $9$, so $-9$ won't work. But $-10$ would simplify to $10$ and that works. So everything below $-9$ would be made positive and be greater than $9$. Therefore, we can graph:

Therefore, we turned the absolute value inequality into a compound inequality: $$x<-9\quad\text{or}\quad x>9$$

Next up, Patrick looks at the inequality: $|x|\le 6$

To turn this absolute value inequality into a compound inequality, again, we start with the positive numbers (and zero). We include $0$ through $5$ since they're all less than $6$. We also include $6$ since we have a less than or equal to sign. We do not include $7$ since it's bigger than $6$.

Next, look at the negative numbers. Notice that $-1$ through $-5$ simplify to $1$ through $5$ respectively. These are all less than $6$. Likewise, $-6$ simplifies to $6$, which we include since the sign is less than or equal to. Finally, $-7$ simplifies to $7$, which we don't include since it's greater than $6$. We can graph:

So, we turned the absolute value inequality into a compound inequailty: $$-6\le x\le6$$

The correct answer is $|x| \le 3.$ All of the numbers from $0$ to $3$ are filled in, since they are all less than or equal to $3$ and this is an inclusive inequality. All of the numbers from $-3$ to $0$ are filled in because they are less than or equal to $3$ when you take the absolute value of them.

Good job using your skills to turn a compound inequality back into an absolute value inequality!

Turning Absolute Value Inequalities Into Compound Inequalities

Translate absolute value inequalities into compound inequalities to solve. If the absolute value is greater than a number, use an OR inequality. If it's less, use an AND inequality.

Remember absolute value? As a quick review, when you take the absolute value of a number, the result is the non-negative version of the number. For example,

$$|-5| = 5\\ |5| = 5, \text{ since } 5 \text{ is already non-negative}\\ |0| = 0, \text{ since } 0 \text{ is already non-negative}$$

An absolute value inequality is an inequality involving an absolute value. We can split these inequalities into compound inequalities.

Let's look at some examples of solving the inequality.

$$\textbf{Turning Absolute Value Inequalities into AND Compound Inequalities}$$

Consider the following absolute value inequality:

$$|x| < 6$$

Say $x$ is a non-negative number. Since the absolute value sign does not affect non-negative numbers, let's look at the inequality without it.

$$\boxed{x < 6}$$

This is the first part of our compound inequality! Easy! But what happens if $x$ is a negative number?

If $x$ is negative number, we can't just drop the absolute value sign and be done with it! Let's investigate further what happens when $x$ is negative.

Say $x = -1$. Then, $|x| = |-1| = 1$. Indeed, $1<6$.

Say $x = -2$. Then, $|x| = |-2| = 2$. Indeed, $2<6$.

Say $x = -3$. Then, $|x| = |-3| = 3$. Indeed, $3<6$.

Say $x = -4$. Then, $|x| = |-4| = 4$. Indeed, $4<6$.

Say $x = -5$. Then, $|x| = |-5| = 5$. Indeed, $5<6$.

Say $x = -6$. Then, $|x| = |-6| = 6$. But $6$ is not less than $6$.

Say $x = -7$. Then, $|x| = |-7| = 7$. But $7$ is not less than $6$.

And so on.

When $x$ is negative, the only values of $x$ that satisfy the inequality are ones that are greater than $-6$.

$$\boxed{x > -6}$$

Otherwise, when we take the absolute value of $x,$ the inequality is not upheld!

Now, let's put our two discoveries together. We know when $x$ is non-negative, then $\boxed{x < 6}$, and when $x$ is negative, then $\boxed{x > -6}$. If we make this into a compound inequality, we get:

$$-6 < x < 6$$

This is an AND compound inequality. Whenever you have an absolute value inequality of the form

$$|x| < a$$

where $a$ is a non-negative constant, you can make it into an AND compound inequality that looks like this:

$$-a < x < a$$

This works for $\leq$ inequalities too!

$$\textbf{Turning Absolute Value Inequalities into OR Compound Inequalities}$$

Consider the following absolute value inequality:

$$|x| > 7$$

If $x$ is a non-negative number, the absolute value sign does not affect it, so we have: $$\boxed{x > 7}$$

This is the first part of our compound inequality.

Now let's say $x$ is negative number. Let's plug some numbers in for $x$ and see what happens.

Say $x = -1$. Then, $|x| = |-1| = 1$. But $1$ is not greater than $7$.

Say $x = -2$. Then, $|x| = |-2| = 2$. But $2$ is not greater than $7$.

Say $x = -3$. Then, $|x| = |-3| = 3$. But $3$ is not greater than $7$.

Say $x = -4$. Then, $|x| = |-4| = 4$. But $4$ is not greater than $7$.

Say $x = -5$. Then, $|x| = |-5| = 5$. But $5$ is not greater than $7$.

Say $x = -6$. Then, $|x| = |-6| = 6$. But $6$ is not greater than $7$.

Say $x = -7$. Then, $|x| = |-7| = 7$. But $7$ is not greater than $7$.

Wait, last time we had valid inequalities by now! Why isn't it working this time?

For this example, we need to go past $-7$ in order to get a valid inequality.

Say $x = -8$. Then, $|x| = |-8| = 8$. Indeed, $8>7$.

Say $x = -9$. Then, $|x| = |-9| = 9$. Indeed, $9>7$.

Say $x = -10$. Then, $|x| = |-10| = 10$. Indeed, $10>7$.

And so on.

When $x$ is negative, the only values of $x$ that satisfy the inequality are ones that are less then than $-7$.

$$\boxed{x < -7}$$

Now, let's put our two inequalities together. We know when $x$ is non-negative, then $\boxed{x > 7}$, and when $x$ is negative, then $\boxed{x < -7}$. If we make this into a compound inequality, we get:

$$x < -7 \text{ OR } x>7$$

This is an OR compound inequality! For future reference, whenever you have an absolute value inequality of the form

$$|x| > a$$

where $a$ is a non-negative constant, you can make it into an OR compound inequality that looks like this:

$$x < -a \text{ OR } x>a$$ $$\textbf{So, in summary:}$$

Whenever you have an absolute value inequality of the form:

$$|x| < a$$

where $a$ is a non-negative constant, you can make it into an AND compound inequality that looks like this:

$$-a < x < a$$

Whenever you have an absolute value inequality of the form:

$$|x| > a$$

where $a$ is a non-negative constant, you can make it into an OR compound inequality that looks like this:

$$x < -a \text{ OR } x>a$$

If you get a longer expression within the absolute value signs, you can follow the same steps, but also consider the steps to solve compound inequalities!

The patterns of "and" and "or" inequalities

When we're faced with absolute value inequalities, it can feel daunting at first to figure out how to solve the problem. The good news is that there are two main patterns for when we're solving these types of problems!

You may face problems that are a little more difficult that this basic structure, but this pattern will stay relatively the same when tackling absolute value inequalities.