Solving Absolute Value Equations | Applications of Absolute Value Functions |

Solving Absolute Value Inequalities | More Practice |

Graphs of Absolute Value Functions |

As we saw earlier in the **Negative Numbers and Absolute Value** section, an **absolute value** (designated by | |) means take the positive value of whatever is between the two bars. The absolute value is **always** positive, so you can think of it as the distance from **0**. As an example, $ \left| 3 \right|=3$ and $ \left| {-3} \right|=3$. It’s as simple as that!

(Note that we also address absolute values here in the **Piecewise Functions **section and here in the** Rational Functions, Equations, and Inequalities **section.)

## Solving Absolute Value Equations

**Solving absolute equations** isn’t too difficult; just have to separate the equation into two different equations (once we isolate the absolute value), since we don’t if what’s inside the absolute value is **positive** or **negative**. Then, make the expression on the **right-hand side** (without the variables) **both positive and negative** and solve each equation; typically, we will get **two answers**. We must **check our answer****s**, since we may get **extraneous solutions (solutions that don’t work).**.

There are a few cases with absolute value equations or inequalities where we don’t have to even solve! One is when we have isolated the absolute value, and it is set equal to a **negative number**, such as $ \left| {x-5} \right|=-4$, or $ \left| {x-5} \right|\le -4$, for example. Since an absolute value can **never be negative**, we have **no solution** for this case. The other is when the absolute value is greater than a negative number, such as $ \left| {x-5} \right|>-4$ for example. In this case our answer is **all real numbers**, since an absolute value is always positive.

Note that we can always **solve absolute value equations and inequalities graphically**, as shown below.

Here are some problems:

Here’s one more that’s a bit tricky, since we have **two expressions** **with absolute value** in it. In this case, we have to **separate in** **four cases**, just to be sure we cover all the possibilities. We then must **check for** **extraneous solutions**, possible solutions that don’t work.

Here’s another way to approach the absolute value problem above, using **number lines**:

## Solving Absolute Value Inequalities

Note that we learned about **Linear Inequalities** here.

When dealing with absolute values and inequalities (just like with absolute value equations), we have to **separate the inequality into two different ones**, if there are any variables inside the absolute value bars.

**First, get the absolute value all by itself on the left (remember to reverse the inequality sign when multiplying or dividing by a negative number). **Now, separate the equations. We get the first equation by just taking away the absolute value sign away on the left. The easiest way to get the second equation is to take the absolute value sign away on the left, and do two things on the right**: reverse the inequality sign**, and **change the sign of everything on the right** (even if we have variables over there).

We also have to think about whether or not to use “**or**” or “**and**” between the two new equations. The way I remember this is that with a $ >\,\text{or}\,\,\ge $ sign, you can remember “gore”: **greater than uses “or”**. With a $ <\,\text{or}\,\,\le $ sign, think “land”: **less than uses “and”**.

**GORE: Greater Than uses OR**

**LAND: Less Than uses AND**

Note that statement with “or” is a **disjunction**, which means that it works if only one (or both) parts are true. A statement with “and” is a **conjunction**, which means it only works if both parts are true.

And again, if we get something like $ \left| {x+3} \right|<0$ (or a negative number), there is **no solution**, and something like $ \left| {x+3} \right|\ge 0$ (or a negative number), there are **infinite solutions** (all real numbers).

Also, remember to use open brackets for inequalities that aren’t inclusive ($ <$ and $ <$) and closed brackets for inequalities that are inclusive and include the boundary point ($ \le $ and $ \ge $).

Here are some examples:

There are examples of **rational functions with absolute values** here in the **Rational Functions, Equations, and Inequalities** section.

## Graphs of Absolute Value Functions

Note that you can put absolute values in your **Graphing Calculator** (and even graph them!) by hitting **MATH, **scroll right to **NUM**, and then hitting **1 (abs) **or** ENTER**.

**Absolute Value functions** typically look like a **V **(upside down if the absolute value is negative), where the point at the **V** is called the **vertex**. For the absolute value parent function, the vertex is at $ \left( {0,0} \right)$.

We looked at **absolute value parent functions and their transformations **in the **Absolute Value Transformations** section, and **absolute value functions as piecewise equations** here in the **Piecewise Functions** section.

Note that the general form for the absolute value function is $ f\left( x \right)=a\left| {x-h} \right|+k$, where $ \left( {h,k} \right)$ is the vertex. If $ a$ is positive, the function points down (like a **V**); if $ a$ is negative, the function points up (like an upside-down **V**). Here’s a graph of the parent function, and also a transformation:

You can **solve absolute value equations and equalities with graphing**; here are some examples of solving inequalities:

## Applications of Absolute Value Functions

Absolute Value Functions are in many **applications**, especially in those involving V-shaped paths and **margin of errors**, or **tolerances**. Here are some examples **absolute value** “word” problems that you may see:

Here are examples that are **absolute value inequality applications**. **Use this rule of thumb**: the absolute value of a difference is usually on the left-hand side, the amount that differs or varies is usually on the right-hand side, with a $ <$ or $ \le $ sign in between.

**Learn these rules, and practice, practice, practice!**

Click on Submit (the arrow to the right of the problem) to solve this problem. You can also type in more problems, or click on the 3 dots in the upper right hand corner to drill down for example problems.

If you click on “Tap to view steps”, you will go to the **Mathway** site, where you can register for the **full version** (steps included) of the software. You can even get math worksheets.

You can also go to the **Mathway** site here, where you can register, or just use the software for free without the detailed solutions. There is even a Mathway App for your mobile device. Enjoy!

On to **Solving Radical Equations and Inequalities** – you’re ready!