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Chapter 12: Problem 39

Graph each system of inequalities. \(y \leq-x^{2}\) \(y \geq x-3\) \(y \leq-1\) \(x \leq 1\)

Short Answer

Expert verified
Graph each inequality on the same coordinate plane and shade the corresponding regions. The overlapping area shows the solution.

Step by step solution

01

Graph the first inequality

Start by graphing the equation of the parabola: \(y = -x^2\). Because the inequality is \(y \leq -x^2\), shade the region below and on the parabola.
02

Graph the second inequality

Next, graph the equation of the line: \(y = x - 3\). Since the inequality is \(y \geq x - 3\), shade the region above and on the line.
03

Graph the third inequality

Graph the horizontal line given by \(y = -1\). Because the inequality is \(y \leq -1\), shade the region below and on the line.
04

Graph the fourth inequality

Finally, graph the vertical line represented by \(x = 1\). Since the inequality is \(x \leq 1\), shade the region to the left and on the line.
05

Find the solution region

Identify the region on the graph where all shaded areas overlap. This common region satisfies all the given inequalities.

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Key Concepts

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

parabola
A parabola is a U-shaped graph that can open upwards or downwards. In this exercise, the given parabola is represented by the equation \(y = -x^2\), which opens downwards. To graph this parabola, plot points where the x-values are equally spaced on both sides of the vertex (the highest point if it opens downward or the lowest if it opens upward). For \(y \leq -x^2\), you'll shade the region below and including the parabola, because any point in this area satisfies the inequality. To visualize this more clearly:
  • The vertex is at (0,0).
  • For points like (1, -1) and (-1, -1), you'll see they fall on the parabola and anything below should be shaded.
linear inequality
A linear inequality involves a linear function and defines a region on a graph. For example, in this problem, \(y \geq x - 3\), you first graph the line \(y = x - 3\). This linear function is a straight line with a slope of 1 and intercepts the y-axis at -3. To graph:
  • Start at (0, -3) on the y-axis.
  • Use the slope (rise over run) to find another point: from the y-intercept, go up 1 unit and right 1 unit to (1, -2).
  • Draw a straight line through these points.
  • Since the inequality is \geq\, shade the area above and on the line.

This region includes all points where y-values are either on or greater than the line values.
horizontal line
A horizontal line in a graph signifies that y remains constant regardless of x. The equation for a horizontal line is written as y = a constant. For instance, in \(y \leq -1\), the line \(y = -1\) is graphed at every point where the y-coordinate is -1. Since the line represents \(y \leq -1\):
  • Find -1 on the y-axis and draw a straight line horizontally through.
  • Shade the region below and on this line because y-value for any point here must be less than or equal to -1.
vertical line
A vertical line shows that x is fixed while y can be any value. This is displayed as x = a constant. In the current task, \(x \leq 1\) means:
  • On the x-axis, locate 1 and draw a vertical line that runs up and down from here.
  • Shade the region to the left of this line, including the line itself.

This region includes all the points where the x-values are either on the line or less than 1.

Bringing all these together by overlapping shaded areas from each inequality gives the solution region satisfying all conditions.

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Most popular questions from this chapter

Write the center-radius form of the circle with the given equation. Give the center and radius, and graph the circle. $$x^{2}+y^{2}-4 x-6 y+9=0$$

Solve each system using the substitution method. \(x^{2}+y^{2}=4\) \(y=x^{2}-2\)

Solve each problem using a nonlinear system. A company has found that the price \(p\) (in dollars) of its scientific calculator is related to the supply \(x\) (in thousands) by the equation $$ p x=16 $$ The price is related to the demand \(x\) (in thousands) for the calculator by the equation $$ p=10 x+12 $$ The equilibrium price is the value of \(p\) where demand equals supply. Find the equilibrium price and the supply/demand at that price. (Hint: Demand, price, and supply must all be positive.)

Graph each inequality. \(y^{2}-16 x^{2} \leq 16\)

Solve each system using the substitution method. \(x y=-3\) \(x+y=-2\)
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