Can some explain very quickly what $ |5 x + 20| = 5 $ actually means?

Question

I don’t mean to bother the community with something so easy, but for the life of me, I can’t remember how to do these. I even forgot what they were called, and I’m referring to the use of the “$ | $” symbol in math. I think I may have been studying too much all day. This is my last section, and if I can’t even remember the name, then I can’t look it up.

Can some please tell me what these types of math problems are called and how to solve them? I just need to see someone do it. I remember it was easy, but it’s been sometime since I’ve last seen these. I really do appreciate everyone helping me study today. I couldn’t of done it without ya! ^_^

Problem. Solve for $ |5 x + 20| = 5 $ for $ x $.

Answer 1

$|\cdot|$ denotes the absolute value here. To solve for $x$, you can break the problem into two cases to get rid of the absolute value.

Consider the cases $5x+20<0$ and $5x+20\geq0$ separately and solve for $x$.

Answer 2

$$|5x+20|=5$$

We want to consider both $5$ and $-5$ for this absolute value function. So we solve for both $5$ and $-5$.

$$5x+20=5$$ $$5x=-15$$ $$\boxed{x=-3}$$ $$5x+20=-5$$ $$5x=-25$$ $$\boxed{x=-5}$$

If we plug $-3$ and $-5$ back into the original equation, we should get an equivalent statement. So the final answer is $\boxed{x\in \{-5,-3\}}$.

If you would like more information on solving absolute value equations and inequalities, please check this out.

Answer 3

The equation $$\lvert a+b\rvert=c$$ may be read as "the distance from the number $a$ to the number $-b$ is $c$."

With this in mind, the equation $$ \lvert5x+20 \rvert=5 $$ is the collection of all $x$ so that the distance from the number $5x$ to the number $-20$ is $5$. Can you find all such $x$?

Answer 4

As an alternative using $|a|=\sqrt{a^2}$ $$ \begin{align} |5x+20|&=5\\ \sqrt{(5x+20)^2}&=5 \\(5x+20)^2&=25 \\ 25x^2+200x+400&=25\\ x^2+8x+15&=0 \\(x+3)(x+5)&=0 \\ x =-3,-5\end{align}$$

Absolute value is sometimes called Modulus

I thought I'd upload a picture whilst I was using Geogebra. The intersections are shown.