Final answer:
To find the domain in an inequality with fractions, simplify the fractions, identify the variables and their restrictions, find the valid ranges, and write the domain using interval notation or set notation.
Step-by-step explanation:
The domain of an inequality with fractions on either side can be determined by considering the restrictions on the variables involved. To find the domain, we need to look for values of the variable that make the inequality true. Here are the general steps to find the domain in an inequality with fractions:
- Simplify the fractions on both sides of the inequality, if possible.
- Identify any variables involved in the inequality.
- Consider any restrictions on the variables. For example, if a variable appears in the denominator of a fraction, check if it can be equal to zero.
- Find the valid ranges of the variables that satisfy the inequality based on the restrictions.
- Write the domain using interval notation or set notation.
Here's an example:
If we have the inequality (x - 2) / 3 < 2 / (x + 3), we can start by simplifying the fractions: (x - 2) / 3 < 2 / (x + 3).
Next, we identify the variable x and consider any restrictions. In this case, the variable x cannot be equal to -3, because that would make the denominator of the second fraction equal to zero.
Then, we find the valid ranges of x that satisfy the inequality. We can solve the inequality algebraically or graphically to determine the valid ranges. In this example, we can solve the inequality algebraically to find that the valid ranges for x are x > -3 and x < 2.
Finally, we can write the domain using interval notation or set notation: x > -3, x < 2 or (-3, 2).