91Ó°ÊÓ

Linear inequalities are similar to linear equations, but instead of an equal sign, they use inequality symbols like \(<, >, \leq, \geq\). Linear inequalities represent a range of possible values for a variable rather than a single value. In our example, the original inequality is \(2x - 7 \leq 21\). This means that the left-hand side expression, \(2x - 7\), can be equal to 21 or any value less than 21. In linear inequalities, you use similar steps to solving a linear equation like isolating the variable, but you must also pay attention to the direction of the inequality sign, especially when multiplying or dividing by negative numbers, as this can reverse the direction of the inequality.

Interval notation is a convenient way of expressing the set of solutions to an inequality. It conveys the range of possible values neatly and concisely. For inequalities like \(x \leq 14\), the set of solutions includes numbers that are less than or equal to 14. In interval notation, this is expressed as \((-\infty, 14]\). Here's how it works:

• The round bracket \((\) is used when the endpoint is not included in the interval, which is the case for negative infinity.
• The square bracket \([\) indicates that the endpoint is included, which is used with 14 in our solution.
• Negative infinity \((-\infty)\) is always used with a round bracket because infinity is not a specific number and cannot be "reached."

Solving inequalities involves finding the values of a variable that make the inequality statement true. The general process is much like solving equations:

• Isolate the variable: Start by using algebraic operations to get the variable alone on one side of the inequality. In our exercise, adding 7 to both sides helped to isolate the linear term \(2x\).
• Perform necessary operations: Next, divide or multiply as needed to completely isolate the variable. Divide both sides by 2 to solve \(2x \leq 28\), resulting in \(x \leq 14\).
• Consider inequality direction: Remember when multiplying or dividing by a negative number, it's crucial to reverse the inequality sign. This is an essential step to keep in mind, even though it wasn't needed in our example.
• Express the solution: Finally, express the solution in an appropriate format, like interval notation, which shows all possible solutions as \((-\infty, 14]\).

Following these steps will guide you through solving most basic linear inequalities effectively.