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Chapter 7: Problem 78

Graph the solution set of each system of inequalities by hand. $$\begin{array}{l}y>\frac{1}{x^{2}} \\\y>x^{2}\end{array}$$

Short Answer

Expert verified
The solution set is the region above both the curve \(y = \frac{1}{x^2}\) and the parabola \(y = x^2\).

Step by step solution

01

Review Inequalities

The given system of inequalities consists of two inequalities: 1. \( y > \frac{1}{x^2} \)2. \( y > x^2 \)These need to be graphed on the same coordinate plane to find the solution set that satisfies both conditions simultaneously.
02

Graph \( y = \frac{1}{x^2} \)

Graph the equation \( y = \frac{1}{x^2} \) as a dashed line because the inequality is strictly greater than. This curve is a hyperbola opening upwards, with vertical asymptotes at \( x = 0 \) and approaches the x-axis as \( x \rightarrow \infty \).
03

Identify Region for \( y > \frac{1}{x^2} \)

Since the inequality is \( y > \frac{1}{x^2} \), shade the region above the hyperbolic curve from Step 2. This area represents all the points where the y-values are larger than those on the curve.
04

Graph \( y = x^2 \)

Graph the equation \( y = x^2 \) as a dashed line because the inequality is strictly greater than. This forms a parabola that opens upwards, with the vertex at the origin (0,0).
05

Identify Region for \( y > x^2 \)

Since the inequality is \( y > x^2 \), shade the region above the parabola from Step 4. This area represents all the points where the y-values are greater than those on the parabola.
06

Combine Shaded Regions

The solution to the system is the region where the shaded areas from Steps 3 and 5 overlap. This is the area above both the hyperbolic graph and the parabolic graph.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Systems of Inequalities
A system of inequalities is a set of two or more inequalities that are considered together. The goal is to find a common solution set that satisfies all inequalities in the system. In our specific exercise, we deal with the inequalities \( y > \frac{1}{x^2} \) and \( y > x^2 \).
When graphing systems of inequalities:
  • Each inequality is treated separately first, plotted as if equalities (using dashed lines for strict inequalities like \( > \) or \( < \)).
  • The solution set for each inequality is represented by shading the region that satisfies the inequality condition.
  • The solution for the overall system is the intersection of the shaded regions that satisfy all inequalities.
Combining these steps graphically highlights the intersecting area where all conditions hold true, giving us the overall solution region.
Hyperbola
A hyperbola is a type of curve on a graph that looks somewhat like two mirrored arcs. Unlike parabolas, hyperbolas can open in different directions depending on their equations.
For the inequality \( y > \frac{1}{x^2} \), we graph the equation \( y = \frac{1}{x^2} \). This forms a hyperbola with a vertical orientation. Here are some characteristics of this hyperbola:
  • It has vertical asymptotes along the y-axis at \( x = 0 \), meaning the graph approaches but never touches the y-axis.
  • The branches of this hyperbola become close to the x-axis as \( x \rightarrow \infty \), providing an upward-opening shape.
To solve \( y > \frac{1}{x^2} \), we shade the area above the hyperbola because we need all y-values greater than \( \frac{1}{x^2} \).
This shaded area represents all potential solutions satisfying this inequality.
Parabola
A parabola is a curved graph representing the quadratic equation \( y = ax^2 + bx + c \). In our case, for the inequality \( y > x^2 \), we plot the parabola \( y = x^2 \).
The parabola in this example:
  • Has a vertex at the origin point (0, 0).
  • Opens upwards, extending infinitely in both positive and negative x-directions.
For the inequality \( y > x^2 \), we are interested in the area above the parabola, signifying y-values that exceed those on the parabola. Hence, we shade the region above the parabolic curve.
The overlap between the shaded regions from both the parabola and hyperbola represents the solution set for the system where both inequalities \( y > \frac{1}{x^2} \) and \( y > x^2 \) are true.

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Most popular questions from this chapter

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&x+y \leq 4\\\&\begin{array}{c}x-y \leq 5 \\\4 x+y \leq-4\end{array}\end{aligned}$$

Solve each system by using the matrix inverse method. $$\begin{aligned} &2 x-3 y=10\\\ &2 x+2 y=5 \end{aligned}$$

Graph the solution set of each system of inequalities by hand. $$\begin{aligned}&e^{-x}-y \leq 1\\\&x-2 y \geq 4\end{aligned}$$

Graph each inequality. $$y<\frac{1}{x^{2}+1}$$

Use Cramer's rule to solve each system of equations. If \(D=0,\) use another method to complete the solution. $$\begin{array}{l}x+y+z=4 \\\2 x-y+3 z=4 \\\4 x+2 y-z=-15\end{array}$$
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