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Solving inequalities is a fundamental mathematical skill, similar to solving equations, but with a few key differences. Inequalities tell us how numbers compare to each other, rather than saying they are exactly equal. To solve inequalities, follow these steps:

1. **Breaking down the inequality**: For compound inequalities like the one given, split it into two separate inequalities. For example, when given the compound inequality \(4 \leq 2x + 2 \leq 10\),

we break it into:
- \(4 \leq 2x + 2\)
- \(2x +2 \leq 10\)

2. **Solving each inequality**: Next, treat each part like an equation. For example, with \(4 \leq 2x + 2\), solve by isolating \(x\):
- Subtract 2 from both sides: \(4 - 2 \leq 2x\): Simplifying to \(2 \leq 2x\)
- Divide by 2: \(2/2 \leq 2x/2\): Which simplifies to \(1 \leq x\).

Apply similar steps to solve the other part of the compound inequality. Thus:
- Subtract 2 from both sides: \(2x +2 \leq 10 - 2\): Which simplifies to \(2x \leq 8\)
- Divide by 2: \(2x/2 \leq 8/2\): Which simplifies to \(x \leq 4\).

3. **Combining results**: Once you have solved each part, combine the results to give the overall solution:
\(1 \leq x \leq 4\).

This means that \(x\) must be greater than or equal to 1 but less than or equal to 4.

Interval notation helps to express sets of numbers using intervals. This notation is a convenient way to describe solutions to inequalities. Here's how:

- **Square brackets [ ]**: Indicate that the endpoints are included in the interval (known as closed intervals).

- **Parentheses ( )**: Indicate that the endpoints are not included in the interval (known as open intervals).

To express the solution of our inequality \(1 \leq x \leq 4\) in interval notation, we use square brackets, since both endpoints - 1 and 4 - are included:

The interval notation for this solution is \[1, 4\].

How to read this:

• **[1, 4]**: x includes values starting from 1 up to and including 4.

Using interval notation simplifies expressing complex inequalities, especially when dealing with larger ranges or multiple conditions.

Graphing inequalities visually represents the solution set on a number line. This provides a clear, visual interpretation of the solution. Let's learn how to graph the solution \[1, 4\]:

1. **Plotting critical points**: Mark the points 1 and 4 on the number line.

2. **Closed circles**: Because 1 and 4 are included in the solution set (due to 'less than or equal to' and 'greater than or equal to'), place closed circles on these points.

3. **Shading the interval**: Shade the region between 1 and 4. This represents all the values that \(x\) can take.

Graphically, this means:

• A closed circle at 1.
• A closed circle at 4.
• Shading the section of the number line between these two points.

This simple visual gives a clear understanding of where the solution lies on the number line. It helps to identify all possible values that satisfy the inequality.