Absolute Value with two expressions.

[stuart4512]

How do I do this? I have tried a few methods and end up getting x values that don't work when placed back into the equation.

View attachment 3670

 

[MarkFL]

I would write the expression as a piecewise function:

\(\displaystyle f(x)=\begin{cases}-3x-5, & x<-1 \\[3pt] x-1, & -1\le x\le2 \\[3pt] 3x-5, & 2<x \\ \end{cases}\)

Can you proceed?

edit: I have moved this thread here from our Linear Algebra subforum, as this is a better fit. :D

 

[stuart4512]

I got the only solution to be at x=1.
How do I sketch the area between -1 and 2?

 

[MarkFL]

I got the only solution to be at x=1.
How do I sketch the area between -1 and 2?

You should find 2 valid roots. Check the root of each piece, and if it is in the given domain for that piece, then it is a valid root.

As for the middle piece, just compute the end points, and connect them with a line segment.

\(\displaystyle f(-1)=-2\)

\(\displaystyle f(2)=1\)

So, plot the points $(-1,-2),\,(2,1)$ and connect them.

 

[CellCoree]

I understand your frustration with trying to solve absolute value equations. It can be a tricky concept to grasp, but with the right approach, it can be solved accurately.

Firstly, it is important to understand that absolute value represents the distance of a number from zero on a number line. This means that no matter what number is inside the absolute value bars, the result will always be a positive value.

To solve an absolute value equation with two expressions, you can follow these steps:

1. Rewrite the equation as two separate equations, one with the positive expression and one with the negative expression inside the absolute value bars.

2. Solve each equation separately by isolating the variable on one side of the equation.

3. Check both solutions by plugging them back into the original equation. If they both work, then you have found the correct solutions.

If you are still getting incorrect values for x, it is possible that there may be an error in your calculations. I would suggest double-checking your work and using a calculator if needed.

Additionally, it is important to keep in mind any restrictions on the variable that may exist in the original equation. These restrictions can affect the validity of your solutions.

In conclusion, solving absolute value equations with two expressions requires breaking it down into simpler equations and carefully checking your solutions. With practice and attention to detail, you can successfully solve these types of equations.