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Chapter 13: Problem 15

Graph each inequality. \(9 x^{2}>16 y^{2}+144\)

Short Answer

Expert verified
Graph the hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \) and shade the region outside of it.

Step by step solution

01

Rewrite the inequality

First, rewrite the inequality in standard form: \[ 9x^2 > 16y^2 + 144 \]Subtract 144 from both sides to obtain:\[ 9x^2 - 16y^2 > 144 \]
02

Divide by 144

Next, divide every term by 144 to put the inequality into standard hyperbola form:\[ \frac{9x^2}{144} - \frac{16y^2}{144} > \frac{144}{144} \]Simplify the fractions to get:\[ \frac{x^2}{16} - \frac{y^2}{9} > 1 \]
03

Identify the hyperbola

Recognize that the equation \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \) is the equation of a hyperbola centered at the origin with a horizontal transverse axis.
04

Graph the hyperbola boundary

Plot the hyperbola \( \frac{x^2}{16} - \frac{y^2}{9} = 1 \) by identifying its vertices and asymptotes. The vertices are at \((4, 0)\) and \((-4, 0)\). The asymptotes are the lines: \[ y = \pm \frac{3}{4} x \]
05

Determine the region to shade

Since the inequality is \( \frac{x^2}{16} - \frac{y^2}{9} > 1 \), we need to shade the region outside the hyperbola. Pick a test point outside of the hyperbola, like \((5, 0)\) and verify: \[ \frac{5^2}{16} - \frac{0}{9} = \frac{25}{16} - 0 = \frac{25}{16} > 1 \]This confirms the region outside the hyperbola should be shaded.

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Key Concepts

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

Hyperbola
A hyperbola is a type of conic section formed by intersecting a double cone with a plane in such a way that the plane cuts both halves of the cone. Hyperbolas have two separate branches, extending to infinity on opposite sides. The standard form of a hyperbola aligns with these properties and helps in graphing it easily. For example, the inequality \(\frac{x^2}{16} - \frac{y^2}{9} > 1\) represents a hyperbola with a horizontal transverse axis and centered at the origin. This means the hyperbola opens left and right along the x-axis.
Standard Form
The standard form of a hyperbola's equation is essential for identifying and graphing it. It usually appears as:\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] or \[ \frac{y^2}{b^2} - \frac{x^2}{a^2} = 1 \]. Here, \(\frac{x^2}{a^2}\) and \(\frac{y^2}{b^2}\) define the relationship between the coordinates and the hyperbola's structure. The horizontal or vertical nature of the hyperbola is determined by which term is positive. For instance, in our exercise, the equation \(\frac{x^2}{16} - \frac{y^2}{9} = 1\) shows it is horizontally aligned. Transforming the inequality to this form involves algebraic manipulations like dividing and simplifying terms to achieve the clear \

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