Problem 53 - College Algebra with Corequisite Support

Question

For the following exercises, solve the system of inequalities. Use a calculator to graph the system to confirm the answer. $$ \begin{array}{l} x y<1 \\ y>\sqrt{x} \end{array} $$

Step-by-Step Solution

Answer

The solutions are in the region below \( y = \frac{1}{x} \) and above \( y = \sqrt{x} \).

Step 1: Understand the Inequalities

Examine the given system of inequalities. The first inequality is linear: \( xy < 1 \), and the second one involves a square root: \( y > \sqrt{x} \). We need to find the region where both inequalities are true.

Step 2: Graph the First Inequality

The inequality \( xy < 1 \) can be considered as a region below the hyperbola \( y = \frac{1}{x} \). Plot the hyperbola \( y = \frac{1}{x} \) on a coordinate plane and shade the region below it, as \( y \cdot x < 1 \) indicates solutions below the hyperbola.

Step 3: Graph the Second Inequality

For the inequality \( y > \sqrt{x} \), plot the curve \( y = \sqrt{x} \), which is the right half of a parabola opening upwards. Shade the region above this curve, which represents all the points where \( y \) is greater than \( \sqrt{x} \).

Step 4: Find the Intersection of the Regions

Determine the area where the shaded region from Step 2 (below the hyperbola) and Step 3 (above the square root function) overlap. This common region represents the solutions to the system of inequalities.

Step 5: Confirm with a Calculator

Use a graphing calculator or graphing software to plot both \( y = \frac{1}{x} \) and \( y = \sqrt{x} \). Verify that the shaded region from the calculator matches the manually determined region in Step 4. The safe region lies below the hyperbola and above the parabola.

Key Concepts

Graphing Inequalities

Graphing inequalities involves visualizing the solution sets on a coordinate plane. This can significantly help to understand where a particular mathematical condition holds true. When graphing inequalities,

• Solid lines are used to show boundaries included in the solution (using ≤ or ≥ signs).
• Dashed lines are used where the boundary is not included (using < or > signs).

For the inequality \( xy < 1 \), we graph the hyperbola \( y = \frac{1}{x} \) with a dashed line. This shows that points on the hyperbola itself don't satisfy the inequality.
For the inequality \( y > \sqrt{x} \), the boundary \( y = \sqrt{x} \) is drawn as a dashed line to indicate that points exactly on the boundary are not part of the solution. In both cases, the areas of interest are either the region below (for the hyperbola) or above (for the square root). Successfully graphing these inequalities provides a visual approach to solving the system, helping us find where the two conditions overlap.

Hyperbolas

A hyperbola is a type of conic section that looks like two symmetrical open curves. In mathematics, it’s critical to recognize hyperbolas, as they often appear in equations involving inverse relationships. The standard form of a hyperbola is:\[ y = \frac{a}{x} \]which can also be viewed similar to our example \( y = \frac{1}{x} \).
Key elements of a hyperbola include:

• Asymptotes: These are lines that the curve approaches but never touches. For \( y = \frac{1}{x} \), both the x-axis and y-axis are asymptotes.
• Center: The center acts as a point of reference, typically the origin in simple forms like \( y = \frac{1}{x} \).
• Branches: The hyperbola has two separate curves or branches that can extend infinitely.

Understanding these properties makes it easier to graph and interpret hyperbolas, allowing students to correctly plot inequalities involving them. The solution region for \( xy < 1 \) is found below where these curves form.

Square Root Functions

Square root functions are a particular type of function involving the square root of the input variable. The general form is \( y = \sqrt{x} \). This function graphically forms half of a parabola that opens to the right.
Important aspects of square root functions include:

• Domain: The function is defined for non-negative x-values (\( x \geq 0 \)), as square roots of negative numbers are not real.
• Graph: The graph starts at the origin \((0,0)\) and increases steadily without bound as x increases.
• Curvature: Unlike a straight line, the graph has a curved pathway that initially grows sharply and then levels off.

The critical consideration when graphing \( y > \sqrt{x} \) is recognizing the domain and how it affects the overlap with other graphed inequalities in a system. For instance, the solution for \( y > \sqrt{x} \) lies above the curve, showcasing a region of growth beyond the parabola's typical path.