91Ó°ÊÓ

Solving inequalities involves finding the values of a variable that satisfy given conditions. In this exercise, we tackle a compound inequality, which means that there are two inequalities connected by a logical connection, often 'and'. To solve such inequalities, you should approach each part individually and then combine the results.

Start by inspecting inequalities separately. For example, with an expression like \(3 \leq 4x-3 < 19\), handle each step carefully:

• Add or subtract numbers on all sides to isolate mathematical expressions. Here, we add 3 to each side.
• Subsequently, solve for the variable by dividing all terms by the coefficient in front of 'x'. This gives us a clearer range for the values of 'x'.

You must be cautious with operations. For example, multiplying or dividing by a negative number would reverse the inequality signs. This is crucial for solving inequalities accurately.

A good method for checking your result is to substitute values back into the original inequality to verify if they satisfy both parts. This approach reinforces understanding and ensures accuracy in finding the correct solution range.

Interval notation is a method to describe a set of numbers within a range concisely. It is particularly useful when dealing with inequalities because it succinctly represents the solution set of a variable.

To depict an interval, we use brackets:

• Square brackets \([ \text{ and } ]\) mean that the endpoint is included in the interval. This is known as a closed interval.
• Round brackets ( \(\text{ and } )\) indicate that the endpoint is not included, known as an open interval.

For instance, after solving our compound inequality, we arrived at \([1.5, 5.5)\). This indicates that 1.5 is included in the solution (represented by [ ), but 5.5 is not included (hence the use of )).

This notation compresses information, making it easier to read and understand compared to repeating the inequality itself.

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. In our compound inequality, \(4x - 3\) is a key algebraic expression. Understanding how to manipulate these expressions is vital to solve inequalities effectively.

When handling algebraic expressions in inequalities, make sure to follow these steps:

• Isolate expressions: Adjust the inequality by performing operations to both sides, like adding or subtracting terms, to make expressions easier to manage.
• Balance both sides: Whatever you do to one side of the inequality, must be done to the other(s). This rule maintains the balance and correctness of the inequality.

Algebra often involves simplifying or restructuring expressions to reveal the solution clearly. Practicing these skills will boost your problem-solving expertise and deepen your understanding of algebraic concepts, essential for mastering inequalities.