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A linear inequality is similar to a linear equation, but instead of an equal sign, it has an inequality symbol. It could be less than (<), greater than (>), less than or equal to (\leq), or greater than or equal to (\geq).
When solving inequalities, the goal is to isolate the variable on one side. This example uses the inequality:\[2x - 7 < 11\]It follows a simple two-step process to solve. First, adjustments are made by adding or subtracting terms, and then we multiply or divide to find the solution. Solving inequalities often consists of performing the same mathematical operations on both sides just like in equations. However, remember that if you multiply or divide by a negative number, you must flip the inequality symbol.

• Linear inequalities express a range of possible solutions.
• They require careful handling, especially concerning signs during multiplication or division.

Interval notation is an efficient way to express a range of values, particularly for inequalities. Once you solve an inequality like \(x < 9\), you can represent it using interval notation. In this format:

• Use a parenthesis "(" or ")" to indicate endpoints that are not included.
• Use a bracket "[" or "]" to show endpoints that are included in the solution.

For the solution \(x < 9\), the value of 9 is not included, thus, we express this in interval notation as:\[(-\infty, 9)\]The interval means all values from negative infinity up to, but not including, the number 9. Infinity symbols are always used with parentheses because you cannot actually "reach" infinity.
Interval notation is a concise way to communicate the solution set of inequalities.

Algebraic manipulation involves rearranging mathematical expressions to isolate the variable of interest. Let's go through this systematically with our inequality. Begin with:\[2x - 7 < 11\]
This problem requires basic algebraic techniques that include adding, subtracting, multiplying, and dividing terms.

• First, add 7 to both sides to start eliminating terms not connected to the variable, leading to:\[2x < 18\]
• Then, divide both sides by 2. This gives:\[x < 9\]

These actions ensure shifts are made correctly without altering the inequality's direction.
Algebraic manipulation in the context of inequalities helps simplify complex expressions and makes them easier to solve.