91Ó°ÊÓ

In mathematics, inequalities involve the use of signs like \( >, <, \geq, \leq \) to show one side is larger or smaller than the other. When solving inequalities, the goal is to find all values of a variable that make the inequality true.

For example, consider the inequality \(5x - 3 \geq 2\). To solve it, we aim to isolate \(x\). Start by adding 3 to both sides to cancel out the negative 3, which gives us \(5x \geq 5\).

Now, divide both sides by 5 to isolate \(x\), resulting in \(x \geq 1\). Follow similar steps for the second inequality \(6 \geq 4x - 3\): add 3 to both sides, resulting in \(9 \geq 4x\), then divide by 4 to find \(x \leq \frac{9}{4}\).

When working with compound inequalities, finding a common solution set or the overlap of the solutions is key. Here, both conditions \(x \geq 1\) and \(x \leq \frac{9}{4}\) must be satisfied at the same time. Hence, the solution is the interval \(1 \leq x \leq \frac{9}{4}\).

Interval notation is a concise way to express a range of values, typically used with inequalities. It involves brackets and parentheses to indicate whether endpoints are included or excluded.

In our exercise, the solution set \(1 \leq x \leq \frac{9}{4}\) can be expressed using interval notation as \([1, \frac{9}{4}]\).

The square brackets \([\) mean that the endpoints 1 and \(\frac{9}{4}\) are included in the solution set. If an endpoint was not included, a parenthesis \(()\) would be used instead. For instance, \((1, \frac{9}{4}]\) would indicate 1 is not part of the solution.

Interval notation streamlines the expression and can easily be interpreted on a number line.

Graphing solution sets visually represents the range of solutions for an inequality on a number line. It’s an intuitive method to showcase which values satisfy the given conditions.

To graph \(1 \leq x \leq \frac{9}{4}\), a closed circle is drawn at 1 and \(\frac{9}{4}\) to show that these numbers are included in the solution.

The section between these endpoints is shaded to indicate all numbers within this range are part of the solution set.

• Closed circles represent included endpoints (use square brackets in interval notation).
• Open circles represent excluded endpoints (use parentheses in interval notation).

Graphs make it easier to understand the scope of solutions and confirm that both inequalities overlap correctly in a compound inequality scenario.