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Chapter 6: Problem 73

Use the shading capabilities of your graphing calculator to graph each inequality or system of inequalities. $$\begin{aligned} &y \geq 2^{x}\\\ &y \leq 8 \end{aligned}$$

Short Answer

Expert verified
The solution is the region where \( y \geq 2^x \) and \( y \leq 8 \) overlap.

Step by step solution

01

Identify the inequalities

We have two inequalities: \( y \geq 2^x \) and \( y \leq 8 \). Our task is to graph these on the calculator and find the region they describe together.
02

Set up the first inequality

For the first inequality \( y \geq 2^x \), enter the equation \( y = 2^x \) into your graphing calculator. This is the boundary line, and the inequality indicates we need the region above this curve.
03

Set up the second inequality

For the second inequality \( y \leq 8 \), enter the equation \( y = 8 \) into your graphing calculator. This is a horizontal line, and the inequality indicates we need the region below this line.
04

Use shading to represent inequalities

Use your graphing calculator's shading function. For \( y \geq 2^x \), shade the region above the curve. For \( y \leq 8 \), shade the region below the line.
05

Identify the solution region

The solution to the system of inequalities is where the shaded regions overlap. This is the area that satisfies both conditions simultaneously.
06

Verify the solution

Ensure the overlapping region extends between the curves \( y = 2^x \) and \( y = 8 \). If your graphing calculator can highlight this intersection, use this function to double-check.

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Key Concepts

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

Graphing Calculator
Graphing calculators are powerful tools that can help you visualize mathematical concepts, including inequalities. To start, inputting an equation into a graphing calculator is often done using a specific function or mode designed for graphing functions.
  • Input each inequality as a separate equation. For instance, input \(y = 2^x\) and \(y = 8\) separately to establish the necessary boundary lines.
  • Most graphing calculators have a function to shade areas of interest. Use this to illustrate where the function meets the inequality criterion. For graded areas, indicate which areas meet conditions like \(y \geq 2^x\) or \(y \leq 8\).
Understanding these features helps you effectively visualize and analyze systems of inequalities easily. The calculator doesn't just provide an answer; it displays how those answers look graphically.
System of Inequalities
A system of inequalities involves multiple inequality statements that must be true at the same time. This concept is crucial in fields like linear programming, where multiple constraints must be considered simultaneously.To address such a system:
  • Consider each inequality separately first to understand its implication when graphed.
  • Combine these individual plots to assess where all inequalities are met concurrently. In this example, consider both \(y \geq 2^x\) and \(y \leq 8\).
Visual learners find this graphical method especially useful, as it allows them to see at a glance where all conditions are satisfied. Systems of inequalities often outline regions rather than lines, broadening your perspective on solutions beyond single points.
Shading Regions
Shading is the method used to visually represent solutions on a graph. It indicates where an inequality's conditions are satisfied on the plane. Here's how to think about shading regions:
  • Shading is directional. For \(y \geq 2^x\), shade above the curve, since the inequality signifies greater or equal to values.
  • For \(y \leq 8\), shade below the horizontal line, as all points 'below and on' meet the inequality.
Where these shaded regions overlap, we find the solution to the system of inequalities. Mastering shading helps discover these intersecting areas, assisting in visual solution identification. This technique clarifies not only where but also how inequalities relate on the graph.

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Most popular questions from this chapter

Solve each system graphically. Give \(x\) - and y-coordinates correct to the nearest hundredth. $$\begin{aligned}&y=\sqrt[3]{x-4}\\\&x^{2}+y^{2}=6\end{aligned}$$

Solve each application. Scheduling Production Caltek Computer Company makes two products: computer monitors and printers. Both require time on two machines: monitors, 1 hour on machine \(A\) and 2 hours on machine \(B\); printers, 3 hours on machine \(A\) and 1 hour on machine \(B\). Both machines operate 15 hours per day. What is the maximum number of each product that can be produced per day under these conditions?

Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. Let \(A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right], B=\left[\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right],\) and \(C=\left[\begin{array}{ll}c_{11} & c_{12} \\\ c_{21} & c_{22}\end{array}\right]\) where all the elements are real numbers. Use these matrices to show that each statement is true for \(2 \times 2\) matrices. \(c(A+B)=c A+c B,\) for any real number \(c\)

Graph the solution set of each system of inequalities by hand. $$\begin{array}{r} x+y \leq 36 \\ -4 \leq x \leq 4 \end{array}$$

The relationship between a professional basketball player's height \(h\) in inches and weight \(w\) in pounds was modeled by using two samples of players. The resulting equations were $$\begin{aligned}&w=7.46 h-374\\\&w=7.93 h-405\end{aligned}$$ and Assume that \(65 \leq h \leq 85\) (a) Use each equation to predict the weight to the nearest pound of a professional basketball player who is 6 feet 11 inches. (b) Determine graphically the height at which the two models give the same weight. (c) For each model, what change in weight is associated with a 1 -inch increase in height?
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