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Compound inequalities are like two simpler inequalities combined together. They help us understand ranges of values that satisfy multiple conditions at once.
To solve compound inequalities, we aim to isolate the variable so we know exactly where it lies on the number line. Let’s break it down using our example.
The given inequality is: \( -4 \leq x + 3 \leq 10 \), which can be broken into two simpler inequalities: \( -4 \- x + 3 \leq 10 \).
We isolate the variable by subtracting 3 from each part:
\(-4\ - 3\ \leq x + 3 - 3 \leq 10 - 3 \).
This simplifies to: \( -7 \leq x \leq 7 \).
Now, we understand that x lies between -7 and 7, including both -7 and 7. The goal is always to make one part of the inequality have the variable alone. This makes the solution set clearer and easier to interpret.

Graphing solutions means visually representing the solution of an inequality on a number line.
This helps us see which values satisfy the inequality quickly.
In our example, after isolating the variable, our inequality is: \( -7 \leq x \leq 7 \).
Here’s how we graph it:

• Draw a straight number line.
• Mark points -7 and 7 on the number line.
• Use a closed circle at both -7 and 7 because the inequality is 'less than or equal to' (\( \leq \)). A closed circle shows that the points -7 and 7 are included in the solution.
• Shade the region between -7 and 7. This shaded area represents all the values of x that satisfy the inequality.

By the end, your number line should have a shaded section from -7 to 7, with closed circles at each endpoint. This visual aid makes it clear that any value in this range is a solution to the inequality.

Interval notation is a way of writing the solution set of an inequality using brackets and parentheses.
This method helps describe the set of possible values cleanly and concisely.
For the inequality \( -7 \leq x \leq 7 \):

• We use square brackets \( \[ \] \) to indicate that the endpoints are included. For instance, \[ -7, 7 \] means x can be any number from -7 to 7, inclusive.
• If endpoints were not included (e.g., \( -7 < x < 7 \)), we would use parentheses instead: \(( -7, 7 ) \)
• Always write the smallest value first: here, it’s -7, followed by the largest value, 7.

So, for our example, the interval notation is simply \[ -7, 7 \].
This notation makes it easy to see at a glance the range of values that x can take.