Problem 62 - Algebra and Trigonometry

Question

\(33-66\) . Solve the nonlinear inequality. Express the solution using interval notation and graph the solution set. $$ \frac{x}{2} \geq \frac{5}{x+1}+4 $$

Step-by-Step Solution

Answer

Interval Solution: \((-\infty, -2] \cup [9, \infty)\). Graph this on a number line.

Step 1: Clear the Fraction

Multiply both sides by \(2(x+1)\) to eliminate the fraction. The inequality becomes:\[x(x+1) \geq 10 + 8(x+1)\]

Step 2: Simplify and Rearrange

Expand and simplify both sides:\[x^2 + x \geq 10 + 8x + 8\]Rearrange the terms to bring all terms to one side:\[x^2 + x - 8x - 18 \geq 0\]Simplify further:\[x^2 - 7x - 18 \geq 0\]

Step 3: Solve the Quadratic Equation

To solve \(x^2 - 7x - 18 = 0\), first look for factors:\[(x - 9)(x + 2) = 0\]This gives the roots \(x = 9\) and \(x = -2\).

Step 4: Test Intervals

Test intervals divided by the roots \(-\infty, -2, 9, \infty\):1. Interval \((-\infty, -2)\)2. Interval \((-2, 9)\)3. Interval \((9, \infty)\)Choose a test point from each interval and substitute into \(x^2 - 7x - 18 \geq 0\):- For \(x = -3\): \((-3)^2 - 7(-3) - 18 \geq 0 \rightarrow 9 + 21 - 18 = 12 \geq 0 \)- For \(x = 0\): \(0^2 - 7(0) - 18 \geq 0 \rightarrow -18 < 0\)- For \(x = 10\): \(10^2 - 7(10) - 18 \geq 0 \rightarrow 100 - 70 - 18 = 12 \geq 0\)

Step 5: Write the Solution in Interval Notation

We identified that the inequality \(x^2 - 7x - 18 \geq 0\) is true in the intervals \((-\infty, -2]\) and \([9, \infty)\). Thus, the solution in interval notation is:\[(-\infty, -2] \cup [9, \infty)\]

Step 6: Graph the Solution

Graph the intervals on a number line:- Draw a solid dot at \(-2\) to indicate \(-2\) is included in the solution- Use an arrow pointing to the left from \(-2\) to suggest all numbers less than \(-2\)- Draw a solid dot at \(9\) and an arrow pointing to the right from \(9\) to represent numbers greater than or equal to \(9\).

Key Concepts

Quadratic Inequalities

When we talk about quadratic inequalities, we're dealing with expressions that involve a variable squared, such as \(x^2 - 7x - 18 \geq 0\). These inequalities mean we are finding the range of values for \(x\) that make the expression true.
To solve, you follow these steps:

• Move all terms to one side of the inequality to form \(f(x) \geq 0\), with zero on the other side.
• Rewrite as a quadratic equation, and factor it if possible.
• Identify the points where the expression equals zero, also called roots or zeros.

For example, in the equation \(x^2 - 7x - 18 = 0\), factoring gives \((x - 9)(x + 2) = 0\). The roots, \(x = 9\) and \(x = -2\), help divide the number line into sections where you can test which intervals satisfy the inequality \(\geq 0\). Once you test, you find the intervals that belong to the solution.

Interval Notation

Once you've solved a quadratic inequality, it's time to express the solution in interval notation. This way of writing solutions is concise and shows exactly where the inequality is true. For the inequality \(x^2 - 7x - 18 \geq 0\), the solution \((-\infty, -2] \cup [9, \infty)\) tells us the parts of the number line fulfilling the inequality.
Here's what each part means:

• \((-\infty, -2]\): "from negative infinity to include \(-2\)". The bracket \([]\) indicates \(-2\) is part of the solution.
• \([9, \infty)\): "from \(9\) to positive infinity". The bracket \([]\) indicates \(9\) is part of the solution.
• The union symbol \(\cup\) combines these two intervals indicating that any value in either satisfies the inequality.

This notation provides a simple yet powerful way to communicate where your inequality holds true.

Number Line Graphing

Graphing on a number line is a useful visual method to check solution sets. It helps in understanding where a quadratic inequality is satisfied. For \(x^2 - 7x - 18 \geq 0\), you plot the points \(-2\) and \(9\) on the line.
To indicate which points are part of the solution, follow these rules:

• Use a solid dot at \(-2\) to signal that \(-2\) is included (since it's \(\leq\)).
• Draw an arrow extending left from \(-2\) to show that all values less than \(-2\) are included.
• Use a solid dot at \(9\) to signal that \(9\) is included.
• Draw an arrow extending right from \(9\) to show all values greater than \(9\) are included.

This graph is a great visual reference that aligns with the interval notation \((-\infty, -2] \cup [9, \infty)\), and helps reinforce how the solution behaves in real numbers.