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Chapter 10: Problem 41

Graph each inequality. $$ y \geq-|x|+2 $$

Short Answer

Expert verified
Graph two lines: \(y = -x + 2\) and \(y = x + 2\), then shade above them.

Step by step solution

01

Identify the inequality type

The inequality provided is in the form of a linear inequality with an absolute value, which suggests we will be graphing a region either above or below a boundary line.
02

Rewrite the absolute value inequality as two linear inequalities

The given inequality is \[y \geq -|x| + 2\] The expression \(-|x|\) can be rewritten in two parts as \(-x\) and \(x\). Therefore, we can split the inequality into:1. \[y \geq -x + 2\] 2. \[y \geq x + 2\] These represent the two parts of the line forming a V-shape from the absolute value function.
03

Graph the boundary lines

For the inequality \(y \geq -x + 2\):- The line \(-x + 2\) can be graphed with a starting point (0, 2) and a slope of -1.For the inequality \(y \geq x + 2\):- The line \(x + 2\) can be graphed with the same starting point (0, 2) but a slope of 1.Both lines will meet at the vertex point (0, 2), forming a V-shape.
04

Shading the region

Since the inequalities are \(y \geq -x + 2\) and \(y \geq x + 2\), we shade the region above both lines. This shaded area represents all points (x, y) where y is greater than or equal to values calculated by these lines.

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Key Concepts

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

Graphing Inequalities
Graphing inequalities can seem a little daunting, but it's quite manageable once you understand the basic steps. When graphing an absolute value inequality like \(y \geq -|x| + 2\), we're essentially working with the idea of a boundary line and a region above or below this line.

An absolute value function typically forms a V-shape. For our specific problem, this includes the two linear expressions \(-x + 2\) and \(x + 2\), which are reflections of each other. The task is to graph these two lines on a coordinate plane.

Start by plotting the boundary line for each part of the inequality. When drawing boundary lines, remember:
  • An inequality with \">=" or \

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Most popular questions from this chapter

A ball is thrown straight upward from the top of \(a=\) building with an initial velocity of 32 feet per second. The equation \(s=-16 t^{2}+32 t+48\) gives the height \(s\) of the ball in feet \(t\) seconds after it is thrown. Find the maximum height reached by the ball and the time it takes for the ball to hit the ground.

Solve each inequality. Write the solution set in interval notation and graph it. $$ x^{2} \geq 9 $$

Solve each equation. Approximate the solutions to the nearest hundredth when appropriate. $$ (5 x-2)^{2}=64 $$

CHALLENGE PROBLEMS A. Solve: \(\frac{1}{x}<1\) B. Now incorrectly "solve" \(\frac{1}{x}<1\) by multiplying both sides by \(x\) to clear it of the fraction. What part of the solution set is not obtained with this incorrect approach?

The cost \(C\) in dollars of operating a certain concrete-cutting machine is related to the number of minutes \(n\) the machine is run by the function \(C(n)=2.2 n^{2}-66 n+655\) For what number of minutes is the cost of running the machine a minimum? What is the minimum cost?
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