Problem 7 Graph the solution set of each s... [FREE SOLUTION]

Chapter 13: Problem 7

Graph the solution set of each system of inequalities. a) \(\left[\begin{array}{l}y-x+12\end{array}\right.\) b) \(\left[\begin{array}{l}x^{2}+y^{2} \leq 25 \\ y \geq|x|\end{array}\right.\) c) \(\left[\begin{array}{l}x y<12 \\ x^{2}+y^{2}<16\end{array}\right.\)

Short Answer

Expert verified

The solution to each system of inequalities is the overlapping area of the respective shaded regions of each inequality on the graph.

Step by step solution

Graphing Inequalities for System a)

Start by graphing the curves \(y=x^{2}+8\) and \(y=-x+12\) separately. For \(y-x+12\), shade the region above the line. The solution is where both shaded regions overlap.

Graphing Inequalities for System b)

Begin by graphing the circle \(x^{2}+y^{2}=25\) and the lines \(y=x\) and \(y=-x\). For \(x^{2}+y^{2} \leq 25\), the solution includes the circle and the area inside, so shade this region. For \(y \geq|x|\), which represents the lines \(y\geq x\) and \(y \geq -x\), shade the area above these lines. The solution set is where both shaded areas coincide.

Graphing Inequalities for System c)

Plot first the line corresponding to when the equation \(xy=12\) is met, this is an hyperbola passing through (12,1),(1,12),(-1,-12),(-12,-1). Then, plot the circle given by \(x^{2}+y^{2}=16\). For \(x y<12\), you will shade the area not enclosed by the hyperbola. For \(x^{2}+y^{2}<16\), the solution includes the circle and the area inside it. Therefore, shade this region. The solution set is the intersecting area of these two shaded regions.

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Key Concepts

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

Graphing Inequalities

To graph inequalities, start with each inequality's corresponding boundary line or curve. These boundaries help determine which areas on the graph relate to the inequality sign (\(<, >, \leq, \geq\)).

For example, with the inequality \(y < x^2 + 8\), you begin by plotting the boundary line, which is \(y = x^2 + 8\). This is a parabola that opens upwards.
Next, since we have a \(<\) sign, we shade the region below this curve.
Similarly, for \(y > -x + 12\), plot the line \(y = -x + 12\) and shade above it, because of the \(>\) sign.

The solution set of these inequalities is visually determined by the overlapping shaded areas. These represent all the potential points \((x, y)\) that satisfy both inequalities. The precise graphing of these inequalities allows you to visualize the complex relationship between them.

Solution Sets

Solution sets represent the set of points that satisfy all inequalities in a system. Graphically, you can see this by identifying the overlap of shaded areas for each inequality.

For instance, in system a), the solution set is found at the intersection of the shaded regions below \(y = x^2 + 8\) and above \(y = -x + 12\).
In system b), the solution is where the area inside the circle \(x^2 + y^2 \leq 25\) overlaps with the region above the lines \(y = x\) and \(y = -x\).
In system c), the solution is within the overlap of the area within the circle \(x^2 + y^2 < 16\) and under the hyperbola boundary defined by \(xy = 12\).

Identifying these intersection areas is crucial as they visually represent solutions to the systems of inequalities. They help make sense of complex algebraic relationships in a more intuitive way.

Coordinate Geometry

Coordinate Geometry, or analytic geometry, allows us to use the coordinate plane to visually represent algebraic equations and inequalities.
This visualization translates algebraic inequalities into geometric regions:

Circles are represented by equations like \(x^2 + y^2 = r^2\), where \(r\) is the radius. Shading accordingly shows solutions to circle-related inequalities.
Lines are indicated by linear equations such as \(y = mx + c\), and the region above or below the line is shaded based on the inequality sign.
Parabolas and hyperbolas are also represented, highlighting more complex relationships.

Understanding coordinate geometry assists in recognizing how changes in algebraic expressions shift or alter the graphical representation. This profound connection allows you to tackle numerous mathematical problems efficiently, guided by visual insight.

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91影视

Graph the solution set of each system of inequalities. a) \(\left[\begin{array}{l}y-x+12\end{array}\right.\) b) \(\left[\begin{array}{l}x^{2}+y^{2} \leq 25 \\ y \geq|x|\end{array}\right.\) c) \(\left[\begin{array}{l}x y<12 \\ x^{2}+y^{2}<16\end{array}\right.\)

Short Answer

Step by step solution

Graphing Inequalities for System a)

Graphing Inequalities for System b)

Graphing Inequalities for System c)

Key Concepts

Graphing Inequalities

Solution Sets

Coordinate Geometry

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