Problem 113 - A Graphical Approach to College Algebra

Question

Solve each three-part inequality analytically. Support your answer graphically. $$\sqrt{2} \leq \frac{2 x+1}{3} \leq \sqrt{5}$$

Step-by-Step Solution

Answer

\( \frac{3\sqrt{2} - 1}{2} \leq x \leq \frac{3\sqrt{5} - 1}{2} \)

Step 1: Break into two inequalities

Start by breaking the three-part inequality \( \sqrt{2} \leq \frac{2x+1}{3} \leq \sqrt{5} \) into two separate inequalities: 1) \( \sqrt{2} \leq \frac{2x+1}{3} \) and 2) \( \frac{2x+1}{3} \leq \sqrt{5} \).

Step 2: Solve the first inequality

Take the inequality \( \sqrt{2} \leq \frac{2x+1}{3} \). Multiply every term by 3 to eliminate the fraction: \( 3\sqrt{2} \leq 2x+1 \). Subtract 1 from both sides to isolate the term with \( x \): \( 3\sqrt{2} - 1 \leq 2x \). Finally, divide both sides by 2 to solve for \( x \): \( x \geq \frac{3\sqrt{2} - 1}{2} \).

Step 3: Solve the second inequality

Take the inequality \( \frac{2x+1}{3} \leq \sqrt{5} \). Multiply every term by 3: \( 2x+1 \leq 3\sqrt{5} \). Subtract 1 from both sides: \( 2x \leq 3\sqrt{5} - 1 \). Divide by 2 to solve for \( x \): \( x \leq \frac{3\sqrt{5} - 1}{2} \).

Step 4: Combine solutions

Combine the solutions from both steps, we get: \( \frac{3\sqrt{2} - 1}{2} \leq x \leq \frac{3\sqrt{5} - 1}{2} \). This represents the values of \( x \) that satisfy both inequalities.

Step 5: Graphical representation

Graphically, the solution is a closed interval on the number line from \( x = \frac{3\sqrt{2} - 1}{2} \) to \( x = \frac{3\sqrt{5} - 1}{2} \). Both endpoints are included in the solution.

Key Concepts

Solving Inequalities

When solving a three-part inequality, like \( \sqrt{2} \leq \frac{2x+1}{3} \leq \sqrt{5} \), we need to break it into two manageable parts. This means splitting it into two separate inequalities:

• \( \sqrt{2} \leq \frac{2x+1}{3} \)
• \( \frac{2x+1}{3} \leq \sqrt{5} \)

Once we have these, we solve each inequality on its own, similar to solving regular equations, with the goal of isolating \( x \) on one side. Recall that when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign flips.
Breaking a three-part inequality helps us tackle complex problems step-by-step, ensuring each step is clear and the solution is logically structured.

Analytic Solutions

Analytically solving inequalities involves manipulating algebraic expressions to find all possible solutions for \( x \). This requires multiplying, adding, or dividing terms of the inequality to isolate \( x \) on one side.
For instance, in solving \( \sqrt{2} \leq \frac{2x+1}{3} \), we face fractions and roots. We multiply both sides by 3 to eliminate the fraction. This transforms our inequality to \( 3\sqrt{2} \leq 2x + 1 \). Subtract 1 from both sides to simplify: \( 3\sqrt{2} - 1 \leq 2x \). Finally, dividing by 2 provides \( x \geq \frac{3\sqrt{2} - 1}{2} \).
Solving inequalities analytically requires steps that align with basic algebra rules, providing the exact range or interval for which the inequality holds true.

Graphical Representation

Graphically representing inequalities means showing their solution on a number line or graph. For our solution \( \frac{3\sqrt{2} - 1}{2} \leq x \leq \frac{3\sqrt{5} - 1}{2} \), the number line becomes a powerful visual tool.
On the number line, mark the points \( x = \frac{3\sqrt{2} - 1}{2} \) and \( x = \frac{3\sqrt{5} - 1}{2} \). Draw a solid line between these points to indicate that all numbers between them satisfy both parts of the inequality. These two endpoints will be enclosed with solid dots because the signs "\( \leq \)" indicate that solutions are inclusive of the endpoints.
Visually showing such solutions helps verify calculations and gives a clear picture of the range where the solutions lie.

Interval Notation

Interval notation provides a concise way to describe a range of values that satisfy an inequality solution. It uses brackets and parentheses to indicate if endpoints are included or excluded.
In the inequality \( \frac{3\sqrt{2} - 1}{2} \leq x \leq \frac{3\sqrt{5} - 1}{2} \), both endpoints are included. Thus, the interval notation is \( \left[ \frac{3\sqrt{2} - 1}{2}, \frac{3\sqrt{5} - 1}{2} \right] \).

• Brackets \([ ]\) mean the endpoints are included in the solution (closed interval).
• Parentheses \(( )\) would mean the endpoints are excluded (open interval).

Interval notation is efficient, especially as problems scale in complexity, allowing solutions to be communicated quickly and accurately.