Problem 53 - Algebra and Trigonometry

Question

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{4 x}{2 x+3}>2 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, -\frac{3}{2})\).

Step 1: Clear the Inequality of Fractions

Start by subtracting 2 from both sides of the inequality:\[\frac{4x}{2x+3} - 2 > 0\]Next, express the inequality with a common denominator:\[\frac{4x - 2(2x + 3)}{2x + 3} > 0\]This simplifies to:\[\frac{4x - 4x - 6}{2x + 3} > 0\]Thus, the inequality simplifies to:\[\frac{-6}{2x + 3} > 0\]

Step 2: Analyze the Sign of the Fraction

The fraction \(\frac{-6}{2x+3}\) will be greater than 0 when the denominator, \(2x+3\), is negative because the numerator, -6, is already negative.Find when \(2x+3 < 0\):\[2x + 3 < 0 \]Solve for \(x\):\[2x < -3 \]\[ x < -\frac{3}{2} \]

Step 3: Express the Solution in Interval Notation

The solution for the inequality \(\frac{4x}{2x+3} > 2\) is found when \(x < -\frac{3}{2}\). In interval notation, this is expressed as:\[(-\infty, -\frac{3}{2})\]

Step 4: Graph the Solution Set

To graph the solution set, draw a number line. Mark the point \(-\frac{3}{2}\) on the number line with an open circle, since \(\frac{-3}{2}\) is not included in the solution. Shade the area to the left of \(-\frac{3}{2}\) to indicate all the values for \(x\) that satisfy \(x < -\frac{3}{2}\).

Key Concepts

Interval Notation

When dealing with solutions to inequalities, it's important to express them clearly. Interval notation is a compact and effective way to represent a set of numbers. It's like creating a "summary" of your solution on the number line.

For example, if your inequality solution is all numbers less than -1.5, we write this in interval notation as \((-\infty, -\frac{3}{2})\). This notation tells us:

• The set starts at negative infinity, which means there's no lower bound on the endpoint.
• It ends at -1.5 but does not include -1.5 itself. This is why we use a parenthesis instead of a bracket.

Using interval notation helps us communicate the solution neatly and efficiently, which is why it's a favorite tool in mathematics!

Graphing Inequalities

Graphing inequalities is a visual way to understand the solution set of an inequality. It links the algebraic solution to a real-world representation.

To graph an inequality like \(x < -\frac{3}{2}\), you begin by drawing a number line. This line represents all possible values of \(x\). On this number line:

• Locate the critical point, in this case, \(-\frac{3}{2}\).
• Mark this point with an open circle. The open circle indicates that the exact value of \(-\frac{3}{2}\) is not part of the solution.
• Shade the region to the left of \(-\frac{3}{2}\). This shading represents all the values of \(x\) that satisfy \(x < -\frac{3}{2}\).

The shaded region visually represents all the potential solutions to the inequality, making it easy to see which values are included.

Solution Set

The solution set is simply all the values that satisfy the given inequality. For a nonlinear inequality like \(\frac{4x}{2x+3} > 2\), determining the solution set involves a few careful steps.

First, transform the inequality into a manageable form. Once simplified, find what values make the inequality true. In our example, the inequality simplifies to \(x < -\frac{3}{2}\).

But what exactly is the solution set? It's the collection of \(x\) values that make the original inequality true. It can be conveyed as:

• A list or description (e.g., \(x < -\frac{3}{2}\)).
• By using interval notation (e.g., \((-\infty, -\frac{3}{2})\)).
• Graphically on a number line.

Each of these methods provides a different way to communicate which values of \(x\) will work, but they all express the same critical idea: the range of \(x\) that solves \(\frac{4x}{2x+3} > 2\). Having multiple ways to express a solution is useful for deeper understanding and for solving more complex problems efficiently.