# Finding the Solution Set for an Inequality

• Learning_Math
In summary, the conversation involves finding the solution set for an inequality involving absolute values and fractions. The process involves eliminating the absolute values and considering two subcases for when x is less than or greater than -3. The final solution set is (-inf. -3)U(-3, -11/6).
Learning_Math
1. I need to find the solution set for |(3x+2)/(x+3)|>3.

3. When I solve the inequality (3x+2)/(x+3)>3, I get 2>9 which is clearly false. When I solve the inequality (3x+2)/(x+3)> -3, I come with the solution set (-inf. -11/6). My teacher is saying that there the solutotion set is (-inf. -3)U(-3, -11/6).

I just can't figure out how to get to that solution. I can't figure out where that -3 is coming from. In his sparse notes on my assignment, he says there are two subcases for each of the two cases in number 1. Those are when x < -3 and when x > -3. I just cant' figure out how to use these cases.

The first step for solving $$|X| > a$$, for any expression X and number a, is to eliminate the absolute values with this:

$$X < -a \text{ or } X > a$$

If you need to solve an inequality like (this is entirely made up for illustration)

$$\frac x {x+1} > 5$$

\begin{align*} \frac x {x+1} - 5 & > 0 \\ \frac x {x+1} - \frac{5(x+1)}{x+1} & > 0 \\ \frac{x - (5x+5)}{x+1} & > 0 \\ \frac{-4x - 5}{x+1} & > 0 \\ \frac{(-1)(4x+5)}{x+1} & > 0\\ \frac{4x+5}{x+1} & < 0 \end{align*}

I passed from the next-to-last to the last line by multiplying by (-1).

These steps let you avoid the all-to-common problem of multiplying both sides of an inequality by a variable term when you don't know whether it's positive or negative.

Thanks for that setup. I will remember it for future use. I have also been completely overlooking that (3x+2)/(x+3) is undefined when x = -3. So that is how I get (-inf. -3)U(-3, -11/6) instead of just (-inf. -11/6).

## 1. What is an inequality?

An inequality is a mathematical statement that compares two quantities using the symbols <, >, ≤, or ≥. It indicates that one quantity is greater or less than the other.

## 2. How do you solve an inequality?

To solve an inequality, you must isolate the variable on one side of the inequality symbol and leave the constant on the other side. This is done by using inverse operations, just like solving an equation. However, when multiplying or dividing by a negative number, the direction of the inequality symbol must be flipped.

## 3. What is the solution set for an inequality?

The solution set for an inequality is the range of values that make the inequality true. It can be represented using interval notation or as a graph on a number line.

## 4. How do you graph an inequality?

To graph an inequality, first solve for the variable and then plot the points on a number line. Use an open circle for < and > symbols, and a closed circle for ≤ and ≥ symbols. Then shade the appropriate region to represent the solution set.

## 5. Can an inequality have more than one solution?

Yes, an inequality can have infinitely many solutions. This is because there are infinite numbers that can make the inequality true. However, some inequalities may have no solution if the values for the variable do not make the inequality true.

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