Problem 62 - Algebra and Trigonometry

Question

Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{3+x}{3-x} \geq 1 $$

Step-by-Step Solution

Answer

The solution is \([0, 3)\).

Step 1: Move all terms to one side

Start by moving all terms to one side of the inequality. We have \( \frac{3+x}{3-x} \geq 1 \). Subtract 1 from both sides to give \( \frac{3+x}{3-x} - 1 \geq 0 \).

Step 2: Combine terms to a single fraction

Rewrite the inequality by expressing 1 as a fraction \( \frac{3-x}{3-x} \). This gives us \( \frac{3+x}{3-x} - \frac{3-x}{3-x} \geq 0 \), which simplifies to \( \frac{(3+x) - (3-x)}{3-x} \geq 0 \).

Step 3: Simplify the fraction

Simplify the numerator: \((3+x) - (3-x) = 3 + x - 3 + x = 2x\). So the inequality becomes \( \frac{2x}{3-x} \geq 0 \).

Step 4: Find critical points

Identify the critical points by setting the numerator and denominator equal to zero. For the numerator: \(2x = 0\) gives \(x = 0\). For the denominator: \(3-x = 0\) gives \(x = 3\). These critical points divide the real number line into intervals.

Step 5: Test intervals around critical points

Test sign of the expression \( \frac{2x}{3-x} \) in the intervals defined by the critical points: \((-\infty, 0), (0, 3), (3, \infty)\).- For \((-\infty, 0)\), choose \(x = -1\): \( \frac{2(-1)}{3-(-1)} = \frac{-2}{4} < 0\).- For \((0, 3)\), choose \(x = 1\): \( \frac{2(1)}{3-1} = \frac{2}{2} > 0\).- For \((3, \infty)\), choose \(x = 4\): \( \frac{2(4)}{3-4} = \frac{8}{-1} < 0\). The expression is non-negative in \([0, 3)\).

Step 6: Consider endpoints

Evaluate the inequality at the endpoints:- At \(x = 0\), \(\frac{2(0)}{3-0} = 0 \geq 0\), which satisfies the inequality.- At \(x = 3\), the expression \(\frac{2 \times 3}{3 - 3}\) is undefined, so \(x = 3\) is not included in the solution.

Step 7: Express the solution

The solution is the interval where the expression is non-negative, including the appropriate endpoint. The solution in interval notation is \([0, 3)\).

Key Concepts

Interval Notation

In mathematics, interval notation is a way of representing a subset of the real numbers. When we talk about solutions to inequalities, we often use interval notation to describe the set of numbers that satisfy the inequality. It represents a range of continuous numbers, defined by a lower and an upper boundary. This notation uses brackets and parentheses:

• Square brackets, [ ], mean that the endpoint is included in the interval.
• Parentheses, ( ), mean that the endpoint is not included.

For example, in the solution of our inequality, \([0, 3)\) means all numbers from 0 to 3, including 0 but excluding 3. Using endpoints to determine when to include square brackets or parentheses is crucial for expressing the solution accurately.

Critical Points

Critical points play an essential role in solving nonlinear inequalities. They are the values where the numerator and the denominator of the fraction reach zero, thus dividing the number line into distinct intervals. To find these critical points, we set the numerator and the denominator of the fraction to zero separately. For the given inequality, we first solved the numerator \(2x = 0\) yielding \(x = 0\). Next, we solved the denominator \(3-x = 0\), resulting in \(x = 3\). These critical points allow us to identify the borders of the intervals. Knowing where the function's behavior changes helps us test and determine where the inequality holds true.

Numerators and Denominators

Understanding numerators and denominators is the key to solving inequalities like this one. The expression \((\text{fraction}) \), denotes a division of two expressions, where:

• The numerator (top part) determines when the entire expression equals zero.
• The denominator (bottom part) indicates potential points of discontinuity where the expression might be undefined.

For the inequality \(\frac{2x}{3-x} \geq 0\), we calculated the numerator \(2x\) to find when it becomes zero, which is at \(x = 0\). The denominator \(3-x\) becomes zero at \(x = 3\), meaning the entire expression is undefined at this point. Avoiding division by zero is important when analyzing the solutions. Hence, the numerator and the denominator jointly determine the intervals you will explore to find a complete solution.

Testing Intervals

After identifying the critical points, the next step is testing intervals around these points. These intervals help determine where the inequality holds. For our problem, the critical points divided the number line into three intervals:

• \((-\infty, 0)\)
• \((0, 3)\)
• \((3, \infty)\)

To determine which intervals satisfy the inequality \(\frac{2x}{3-x} \geq 0\), we picked test points from each interval.

• For \((-\infty, 0)\), using \(x = -1\), gave a negative value.
• For \(0, 3\), choosing \(x = 1\), resulted in a positive value.
• For \(3, \infty\), selecting \(x = 4\), provided a negative value.

The expression was non-negative only in \([0, 3)\), which indicates that this interval satisfies the inequality. Testing intervals like this offers a methodical approach to verifying the solution's integrity.