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Interval notation is a simple yet important way to represent a range of values that a variable can take. In mathematics, when we solve inequalities, we often use interval notation to express the solution set concisely. This notation uses brackets and parentheses to describe the set of numbers involved.
If we have a solution where the variable can be equal to a number, we use square brackets, such as \([-4, \infty)\), for including \(-4\). If the variable must be greater than or less than the number without being equal to it, we use parentheses. The infinity symbol (∞) always gets a parenthesis, since infinity itself is not a number that can be reached.

• Example: The inequality \(x \geq -4\) is written as \([-4, \infty)\).

This type of notation provides a clear way to understand solutions for inequalities.

Solving linear inequalities is similar to solving linear equations, with one key difference being the handling of inequality signs. The process generally involves isolating the variable on one side to find its possible values that satisfy the inequality.
For example, given the inequality \(6 - 4x \leq 22\), the goal is to solve for \(x\): by moving the constant term \(6\) to the other side and simplifying, we get \(-4x \leq 16\).

• Step 1: Move terms without \(x\) to the other side.
• Step 2: Isolate \(x\) by performing inverse operations (like subtracting or adding) to both sides.

This straightforward approach helps you quickly determine the range of values for \(x\) that make the inequality true, which can then be expressed in interval notation.

One crucial aspect of solving inequalities is the caveat when multiplying or dividing both sides by a negative number. Remember that this action reverses the inequality sign, which is vital to obtaining the correct solution.
In the inequality \(-4x \leq 16\), when we divide both sides by \(-4\), the inequality sign flips, changing from \(\leq\) to \(\geq\). Therefore, we get \(x \geq -4\). This rule applies across all inequalities and ensures that the solution is accurate, especially when negative coefficients are involved.

• Example: Multiplication or division by -7 would also reverse the signs in an inequality.

Keeping this rule in mind is essential for correct handling of negative coefficients in inequalities.