Problem 45 - Precalculus with Limits

Question

In Exercises 33-46, sketch the graph (and label the vertices) of the solution set of the system of inequalities. $$\left\\{\begin{array}{c}{3 x+4 \geq y^{2}} \\ {x-y<0}\end{array}\right.$$

Step-by-Step Solution

Answer

It is important to sketch the graph carefully considering the nature of the inequalities, quadratic and linear, and then look for the region that satisfies both inequalities. This will result in an area bounded by the line \(y=x\) and the curves \(y=\sqrt{3x+4}\), \(y=-\sqrt{3x+4}\).

Step 1: Sketch the intercepts and graph the first inequality

The first inequality is quadratically inequality \(y^2 \leq 3x+4\). To draw this inequality, interpret it as a quadratic function \(y = \sqrt{3x+4}\) and \(y = -\sqrt{3x+4}\) due to the square root nature of the quadratic function. Sketch these functions using their vertex and symmetry properties. Be sure to attempt to identify the lines \(y=\sqrt{3x+4}\) and \(y=-\sqrt{3x+4}\) where \(y^2\leq3x+4\).

Step 2: Sketch and graph the second inequality

Graph the next inequality \(x-y < 0\), which is equivalent to \(y>x\). This inequality gives all the points above the line \(y = x\). Hence, sketch the line \(y=x\). Above this line corresponds to the solutions of the inequality \(y>x\).

Step 3: Identify the common solution region

Graph both inequalities on the same graph. Now, find the intersection area of the two graphs, which represents the solution set for both inequalities. The solution to the system of inequalities is the region that satisfies both inequalities. Hence, the shaded region, bounded by these curves represents the solution to these inequalities.

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic polynomial is compared to a value using inequality signs, such as ">", "<", "≤", or "≥". In our exercise, the quadratic inequality given is:

• \( y^2 \le 3x + 4 \)

To solve and graph this inequality, we need to understand that it defines a region on the Cartesian plane. A common approach is to first solve the related quadratic equation

• \( y = \sqrt{3x + 4} \)

and its negative counterpart

• \( y = -\sqrt{3x + 4} \).

This pair of equations produces the boundaries of the shaded region that represent solutions to the inequality. The solved inequality means we are looking for points \((x, y)\) that lie between these two curves. These will be the points where the square of \( y \) does not exceed the expression on the right side of the inequality. Quadratic inequalities may involve additional steps, such as identifying intercepts and checking intervals, hence offering a broad area of exploration when solving them.

Graphing Inequalities

The process of graphing inequalities helps visualize the solution set on the coordinate plane. For the first inequality,

• \( y^2 \leq 3x + 4 \)

is represented graphically by sketching the curves \( y = \sqrt{3x + 4} \) and \( y = -\sqrt{3x + 4} \). Sketch these boundaries to visually understand where the inequality holds true. The region between these curves will be shaded to show where \( y^2 ≤ 3x + 4 \) holds true.
For the second inequality,

• \( y > x \)

is graphed as the region above the line \( y = x \). Graphing begins with a straight line passing through the origin with a 45-degree angle, representing all points where y equals x.

To solve the system of inequalities, superimpose these regions on a graph. By observing where the shaded areas from each inequality overlap, you identify the solution set, visually marking areas fulfilling all conditions imposed by both inequalities.

Solution Set

The solution set of a system of inequalities represents all ordered pairs \((x, y)\) that satisfy each inequality in the system simultaneously. Begin by graphing each inequality as separate regions. With

• \( y^2 \leq 3x + 4 \)

formatted as two curves, and

• \( y > x \)

as a dividing line, we find where these inequalities intersect.
The solution set is illustrated as the shared or overlapping region of the shaded graphs. For example, the points lie above the line \( y = x \) and between two curves \( y = \sqrt{3x+4} \) and \( y = -\sqrt{3x+4} \).
It's often helpful to test points within this region to confirm they satisfy both inequalities. This visual representation enriches the understanding of how systems of inequalities operate, illustrating solution sets as tangible areas rather than abstract equations.

Vertex and Symmetry Properties

Vertices and symmetry are essential properties in graphing quadratic equations and inequalities. In our exercise, identifying the vertex of the parabola-like structure aids in drawing an accurate graph of

• \( y = \pm \sqrt{3x + 4} \)

The vertex can be found by setting the expression inside the square root to zero, in this case:

• \( 3x + 4 = 0 \)

This gives us \( x = -\frac{4}{3} \), which aids in sketching the structure around this point with symmetry on either side with respect to the y-axis.
Parabolas are symmetric, meaning whatever happens on one side of the axis of symmetry will also occur on the other. Checking for symmetry allows simpler graph sketching since it's only necessary to plot one side if mirrored correctly.
Recognizing these properties facilitates drawing and interpreting complex inequalities, making them essential tools in both manual graphing and computer-aided solutions.