91Ó°ÊÓ

Interval notation is a way of writing subsets of the real number line. It is especially useful when expressing the solution set of inequalities.
In our exercise, the solution to \(-7 < 2 + 3x < 5\) results in \(-3 < x < 1\) after solving the inequality. This can be translated into interval notation as \((-3, 1)\).
Here's a brief explanation of interval notation:
* Use open brackets \( ( \) and \( ) \) to indicate that the endpoints are not included in the set.
* Use closed brackets \( [ \) and \( ] \) to indicate that the endpoints are included in the set.
For instance, the notation \( (a, b) \) means all numbers between \ a \ and \ b \, without including \ a \ and \ b \ themselves.
Similarly, \( [a, b] \) would include \ a \ and \ b \ as part of the solution set.

Solving multi-step inequalities involves breaking down the problem into manageable steps to isolate the variable. This process is similar to solving equations but requires careful attention to the direction of the inequality symbol.
For example, consider the inequality in our exercise: \(-7 < 2 + 3x < 5\).
Follow these steps to solve multi-step inequalities:
1. **Isolate the variable**: Start by getting rid of any constants and coefficients around the variable. For our example, we first subtract \ 2 \ from all sides to get \-9 < 3x < 3\.
2. **Divide or multiply to solve**: Next, divide or multiply each part of the inequality to solve for the variable. In this case, we divide by \ 3\ to get \-3 < x < 1\.
3. **Write the solution set**: In our exercise, we eventually write \-3 < x < 1\ as \((-3, 1)\) in interval notation. This process ensures that each part of the inequality is treated equally, maintaining the true relationship between all parts.

The steps to solve an inequality are straightforward if followed carefully. Each step should be carried out systematically to ensure the solution is correct.
Let's go through the steps in our example \(-7 < 2 + 3x < 5\):
1. **Subtract 2 from all parts**: This step isolates the term with the variable.
\-7 - 2 < 2 + 3x - 2 < 5 - 2\ simplifies to \-9 < 3x < 3\.
2. **Divide by 3**: Dividing each part by \ 3 \ isolates \ x \. This step looks like \ \frac{-9}{3} < \frac{3x}{3} < \frac{3}{3} \. This simplifies to \-3 < x < 1\.
3. **Write in interval notation**: The solution can then be expressed as \((-3, 1)\) in interval notation.
Always remember to perform the same operation to each part of the inequality and keep track of the inequality direction. Also, when multiplying or dividing by a negative number, remember to reverse the inequality signs.