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Chapter 9: Problem 26

Graph the nonlinear inequality. $$25 x^{2}-36 y^{2}+200 x+144 y-644 \geq 0$$

Short Answer

Expert verified
The inequality describes a hyperbola centered at (-4, 2). Graph it with vertices at (-10, 2) and (2, 2), and follow the direction along the x-axis.

Step by step solution

01

Rewrite the Inequality

The given inequality is \( 25x^2 - 36y^2 + 200x + 144y - 644 \geq 0 \). Begin by moving the constant term to the other side of the inequality to isolate the quadratic terms: \( 25x^2 - 36y^2 + 200x + 144y \geq 644 \).
02

Complete the Square for x-Terms

Focus on the x-terms: \( 25x^2 + 200x \). Factor out the coefficient of \( x^2 \), which is 25: \( 25(x^2 + 8x) \). To complete the square, take half of the coefficient of \( x \) (which is 8), square it (\( 4^2 = 16 \)), and add and subtract it inside the parenthesis: \( 25(x^2 + 8x + 16 - 16) \), simplifying to \( 25((x + 4)^2 - 16) \).
03

Complete the Square for y-Terms

Now complete the square for the y-terms: \( -36y^2 + 144y \). Factor out the coefficient of \( y^2 \), which is -36: \( -36(y^2 - 4y) \). To complete the square, take half of \( -4 \), square it (\( (-2)^2 = 4 \)), and add and subtract inside the parenthesis: \( -36(y^2 - 4y + 4 - 4) \), simplifying to \( -36((y - 2)^2 - 4) \).
04

Simplify the Expression

Substitute back the completed squares for x and y terms into the inequality: \( 25((x+4)^2 - 16) - 36((y-2)^2 - 4) \geq 644 \). Simplify this: \( 25(x+4)^2 - 400 - 36(y-2)^2 + 144 \geq 644 \). Combine constants: \( 25(x+4)^2 - 36(y-2)^2 \geq 900 \).
05

Divide by 900 to Standardize

To transform this into the standard form of an ellipse, divide every term by 900: \[ \frac{25(x+4)^2}{900} - \frac{36(y-2)^2}{900} \geq 1 \]. Simplify: \[ \frac{(x+4)^2}{36} - \frac{(y-2)^2}{25} \geq 1 \]. This expression now represents a hyperbola.
06

Graph the Hyperbola

The standard form \[ \frac{(x+4)^2}{36} - \frac{(y-2)^2}{25} = 1 \] describes a hyperbola centered at (-4, 2) with vertices stretched along the x-axis. Using this form, sketch the hyperbola by plotting the center at (-4, 2), right and left vertices at (-4卤6, 2), and asymptotes determined by the slope given by the ratios from the denominators.

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Key Concepts

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

Completing the Square
When tackling nonlinear inequalities, completing the square is a vital method. This technique centers around converting a quadratic expression, like ax虏 + bx, into a perfect square trinomial. Here's how it works in our case:
  • For the x-terms: consider 25x虏 + 200x.
  • Factor out the 25: 25(x虏 + 8x).
  • To complete the square, take half of 8 (from x虏 + 8x), square it (4虏 = 16), and incorporate it: 25(x虏 + 8x + 16 - 16).
  • This simplifies to 25((x + 4)虏 - 16) as x-terms.
Apply the same to y-terms to form a neat square, turning a complex quadratic into an easier-to-handle equation.
Graphing Hyperbolas
When graphing hyperbolas, understanding their fundamental structure is essential. A hyperbola comprises two mirrored curves diverging from a center point, unlike an ellipse or a circle.
  • In our problem, the key form \( \ rac{(x+4)^2}{36} - \ rac{(y-2)^2}{25} = 1 \) identifies a hyperbola with its center at (-4, 2).
  • Hyperbolas stretch along axes dictated by denominators: here, widest along the x-axis.
  • Sketch begins by plotting center and vertices, then drawing asymptotes which guide the hyperbola's shape.
A clear graph helps visualize solutions to nonlinear inequalities, especially distinguishing where inequalities hold true.
Standard Form of a Hyperbola
The standard form of a hyperbola encapsulates all necessary information to graph it. For a hyperbola, this form follows \( \ rac{(x-h)^2}{a^2} - \ rac{(y-k)^2}{b^2} = 1 \).
  • (h, k) represents the center, determined from completed square shifts, in our case (-4, 2).
  • a虏 and b虏 derive from denominators sharing size and direction of the hyperbola.
  • Given equation shows horizontal stretching: a hyperbola's nature arises when x虏 term precedes y虏.
This concise, organized form simplifies finding key points and sketching accurate graphs.
Inequality Transformation
Transforming complex inequalities into simpler forms is crucial for solving and graphing them. The general task is to isolate terms to reformat inequalities.
  • The original inequality was reshuffled, grouping x and y parts.
  • Post completing square procedures, you simplify and standardize forms.
  • Division by a constant consolidates it, making an inequality recognizable as a hyperbola.
Mastery over these transformations allows solving, interpreting, and graphing nonlinear inequalities smoothly, culminating in understanding their region on a graph.

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Most popular questions from this chapter

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