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Chapter 7: Problem 18

In Exercises 7-20, sketch the graph of the inequality. \( (x - 1)^2 + (y - 4)^2 > 9 \)

Short Answer

Expert verified
The graph of the inequality \((x - 1)^2 + (y - 4)^2 > 9\) will be a circle with a dashed line centered at (1,4) with a radius of 3 units. The area outside the circle is shaded to represent the solution to the inequality.

Step by step solution

01

Identify the Coefficients

The given inequality is \((x - 1)^2 + (y - 4)^2 > 9\). By comparing it with the standard equation of a circle, \(h = 1\), \(k = 4\), and \(r^2 = 9\) from which we can deduce that \(r = 3\). Therefore, the center of the circle is at point (1,4) and the radius of the circle is 3 units.
02

Sketch the Graph

First, draw a Cartesian plane. Then locate the center of the circle at coordinate (1,4). From that point, create a circle with a radius of 3 units. Keep in mind that since we have a '>' sign in the inequality, the circle represented on the graph will be dashed and not solid because the points on the circle are not included in the solution to the inequality.
03

Represent the Inequality Solution Area

The inequality is a 'greater than' inequality, meaning the solution lies outside of the circle, not on or inside the circle. To show this on the graph, shade the entire area outside of the circle.
04

Verifying the Solution

To make sure your graph correctly represents the inequality, take a test point from the shaded area and substitute it into the inequality. If the inequality holds true with those coordinates, the solution graph is correct. If not, review the previous steps for possible errors.

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Key Concepts

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

Circle Equations
When you see an equation like \((x - 1)^2 + (y - 4)^2 = 9\), it represents a circle in a coordinate plane. This is the standard form of a circle equation: \((x - h)^2 + (y - k)^2 = r^2\).
This equation gives us two key pieces of information:
  • The center of the circle, which is \((h, k)\), here being \((1, 4)\).
  • The radius \(r\), found by taking the square root of \(r^2\). In our example, \(r^2 = 9\), so \(r = 3\).
Understanding these components lets you quickly sketch or analyze the circle on a graph. Always remember, the terms \((x - h)\) and \((y - k)\) will give you the center coordinates by simply flipping their signs.
Graphing Inequalities
Graphing inequalities like \((x - 1)^2 + (y - 4)^2 > 9\) involves understanding how to represent areas instead of fixed lines or shapes. Here鈥檚 what you do:
  • The circle from the equation \((x - 1)^2 + (y - 4)^2 = 9\) acts as a boundary.
  • The '>' sign in the inequality tells us to exclude the actual circle's boundary and focus on the area outside.
  • To indicate this, use a dashed circle line to show that perimeter points are not included.
  • Shade the region outside this circle to represent all the points \((x, y)\) that satisfy the inequality.
This visual representation makes it easier to solve and interpret problems involving shapes and their respective inequalities.
Coordinate Geometry
Coordinate Geometry, or Analytic Geometry, is a branch of mathematics where we use coordinates to represent geometric shapes like circles. It helps bridge algebraic equations with geometric visuals.Let's break down the steps:
  • Identify key points like the center \((h, k)\) and radius \(r\).
  • Use these to sketch the shape (a circle here) precisely on the coordinate plane.
  • In inequalities, determine where the solution lies (inside or outside such shapes) using the inequality signs.
  • Test points, like checking a random point beyond the circle, help verify if the shading area chosen satisfies the inequality.
Mastering coordinate geometry enables you to translate mathematical expressions into visual plots, aiding comprehension and offering a powerful way to visualize and solve problems.

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Most popular questions from this chapter

(a) Explain the difference between the graphs of the inequality \( x \le -5 \) on the real number line and on the rectangular coordinate system. (b) After graphing the boundary of the inequality \( x + y < 3 \), explain how you decide on which side of the boundary the solution set of the inequality lies.

Fill in the blanks. An ordered pair \( (a, b) \) is a ________ of an inequality in \( x \) and \( y \) if the inequality is true when \( a \) and \( b \) are substituted for \( x \) and \( y \) respectively.

In Exercises 29-34, the linear programming problem has an unusual characteristic. Sketch a graph of the solution region for the problem and describe the unusual characteristic. Find the minimum and maximum values of the objective function (if possible) and where they occur. Objective function: \( z = x + 2y \) Constraints: \( \hspace{1cm} x \ge 0 \) \( \hspace{1cm} y \ge 0 \) \( x + 2y \le 4 \) \( 2x + y \le 4 \)

In Exercises 41-54, sketch the graph and label the vertices of the solution set of the system of inequalities. \( \left\\{\begin{array}{l} 3x + 4 \ge y^2\\\ x - y < 0\end{array}\right. \)

In Exercises 41-54, sketch the graph and label the vertices of the solution set of the system of inequalities. \( \left\\{\begin{array}{l} x - y^2 > 0\\\ x - y > 2\end{array}\right. \)
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