91Ó°ÊÓ

When faced with a compound inequality, you're dealing with a situation involving two inequalities joined by the words "and" or "or." In the given problem, the expression \(2 \leq x+5 < 4\) is a compound inequality of the form "and." This means we're looking for values of \(x\) that satisfy both inequalities simultaneously.

Here's a quick breakdown:

• The expression \(2 \leq x+5\) requires us to find all values of \(x\) that, when 5 is added, are at least 2.
• The expression \(x+5 < 4\) requires us to find all values of \(x\) that, when 5 is added, are less than 4.

We approach them individually, solve them, and then determine the overlap to find our solution set. In this exercise, the solution to \(2 \leq x+5\) is \(x \geq -3\), and the solution to \(x+5 < 4\) is \(x < -1\). A compound inequality using "and" means we take the intersection: \(-3 \leq x < -1\). This gives us the combined range of values that \(x\) can take.

Once you've solved a compound inequality, it's helpful to express the solution in interval notation. This is a concise way to write the range of values that satisfy the inequality.

For our compound inequality solution \(-3 \leq x < -1\), we express it as \([-3, -1)\). The bracket "[" denotes that -3 is included in the solution, while the parenthesis ")" indicates that -1 is not included. Using interval notation can simplify your solution representation and make it more intuitive to interpret and work with.

• A bracket, like \([\) or \(]\), means that the number is part of the solution (inclusive).
• A parenthesis, like \(()\), means that the number is not part of the solution (exclusive).

Interval notation is particularly handy as it allows those working with inequalities to easily understand the bounds and limits of the solution set.

Graphing inequalities on a number line helps to visualize the set of possible solutions. For the compound inequality \(-3 \leq x < -1\), you'll see a clear depiction of where \(x\) lies.

Here's how to graph this solution:

• Start by identifying the boundaries: \(-3\) and \(-1\).
• Place a solid dot at \(-3\) to show it's included in the solution. Think of this dot as saying "yes, include -3."
• Place an open circle at \(-1\) to indicate it's not included. An open circle essentially means "just short of -1."
• Shade the number line between \(-3\) and \(-1\). This shaded region represents all the values that \(x\) can be.

Graphing is not only a visual confirmation of your solution but also provides a tangible way to see the values \(x\) can take on, enhancing your understanding of the inequality's constraints.