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Chapter 9: Problem 39

Graph the solution set of each system of inequalities. $$\begin{aligned}&y \leq(x+2)^{2}\\\&y \geq-2 x^{2}\end{aligned}$$

Short Answer

Expert verified
Graph the regions \( y \leq (x + 2)^2 \) and \( y \geq -2x^2 \) and find their overlap.

Step by step solution

01

Understand the System of Inequalities

The system includes two inequalities: 1. \( y \leq (x + 2)^2 \)2. \( y \geq -2x^2 \). We need to graph these inequalities on the same coordinate plane and find the region where they overlap.
02

Graph the Boundary of the First Inequality

Graph the parabola \( y = (x + 2)^2 \). This parabola opens upwards and has its vertex at the point \(( -2, 0) \). Draw the parabola and shade the region below it to represent \( y \leq (x + 2)^2 \).
03

Graph the Boundary of the Second Inequality

Graph the parabola \( y = -2x^2 \). This parabola opens downwards and has its vertex at the origin, \(( 0, 0) \). Draw the parabola and shade the region above it to represent \( y \geq -2x^2 \).
04

Identify the Solution Set

The solution set is the region where the shaded areas of both inequalities overlap. This will be the area that satisfies both \( y \leq (x + 2)^2 \) and \( y \geq -2x^2 \) at the same time.
05

Verify the Solution

Check a point in the overlapping region to verify it satisfies both inequalities. For example, checking the point (0, 0):1. \( 0 \leq (0 + 2)^2 \rightarrow 0 \leq 4 \) which is true.2. \( 0 \geq -2 \cdot 0^2 \rightarrow 0 \geq 0 \) which is also true.Thus, the solution set is correctly identified.

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Key Concepts

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

Understanding Parabolas
A parabola is a U-shaped curve that you often encounter in algebra, defined by a quadratic equation. When graphing, it's essential to know the vertex, direction of opening, and general shape. In this exercise, there are two parabolas: one opens upwards, and one opens downwards.

  • The first inequality, \( y \leq (x + 2)^2 \), represents an upward-opening parabola. The vertex of this parabola is at point \( (-2, 0) \).
  • The second inequality, \( y \geq -2x^2 \), represents a downward-opening parabola. The vertex of this parabola is at point \( (0, 0) \).

Knowing these vertices is critical because they are the points where the curve changes direction. By sketching these parabolas on a coordinate plane based on their equations and vertices, you provide a visual foundation for solving the system of inequalities.
Shading Regions in Graphs
In a system of inequalities, shading the correct regions is vital. The shaded areas represent all the possible solutions to the inequality.

For the first inequality, \( y \leq (x + 2)^2 \), we shade below the parabola:
  • Start by drawing the parabola \( y = (x + 2)^2 \).
  • Shade the entire region below this curve.

For the second inequality, \( y \geq -2x^2 \), we shade above the parabola:
  • Draw the parabola \( y = -2x^2 \).
  • Shade the entire region above this curve.

The solution to the system requires identifying where these shaded regions overlap on the graph. This overlapping area includes all the points that satisfy both inequalities at the same time.
Identifying Solution Sets
The solution set of a system of inequalities is the region on the graph that satisfies all inequalities involved simultaneously.

To find it, follow these steps:
  • Graph the inequalities individually.
  • Shade the correct regions for each inequality.

The final step is to observe which part of the graph is shaded by all inequalities. This overlapping region is the solution set.

In our exercise:
  • We shaded below the parabola \( y \leq (x + 2)^2 \).
  • We shaded above the parabola \( y \geq -2x^2 \).

After graphing both, the overlapping shaded area is your solution set. Always verify by choosing points from this region and checking if they satisfy both inequalities to ensure the solution is accurate.

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Most popular questions from this chapter

Solve each problem. A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is \(\$ 351 .\) How many of each denomination of bill are there?

Solve each problem. Purchasing costs The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix \(A\) Eggs lour Sugar Shortening Milk \(\left.\begin{array}{l|ccccc}\text { Rolls (doz) } & 1 & 4 & \frac{1}{4} & \frac{1}{4} & 1 \\ \text { Bread (loaf) } & 0 & 3 & 0 & \frac{1}{4} & 0 \\ \text { Cake } & 4 & 3 & 2 & 1 & 1 \\ \text { Pie (crust) } & 0 & 1 & 0 & \frac{1}{3} & 0\end{array}\right]=A\) The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix \(B\) Large Lot Small Lot \(\left.\begin{array}{l|rr}\text { Eggs } & 5 & 5 \\ \text { Flour } & 8 & 10 \\\ \text { Sugar } & 10 & 12 \\ \text { Shortening } & 12 & 15 \\ \text { Milk } & 5 & 6\end{array}\right]=B\) (a) Use matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options. (b) Suppose a day's orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a \(1 \times 4\) matrix, and, using matrix multiplication, write as a matrix the amount of each ingredient needed to fill the day's orders. (c) Use matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day's orders.

Concept Check Write a system of inequalities for which the graph is the region in the first quadrant inside and including the circle with radius 2 centered at the origin, and above (not including) the line that passes through the points \((0,-1)\) and \((2,2)\)

Given \(A=\left[\begin{array}{rr}4 & -2 \\ 3 & 1\end{array}\right], B=\left[\begin{array}{rr}5 & 1 \\ 0 & -2 \\ 3 & 7\end{array}\right],\) and \(C=\left[\begin{array}{rrr}-5 & 4 & 1 \\ 0 & 3 & 6\end{array}\right],\) find each product when possible. $$A B$$

Find the equation of the circle passing through the given points. $$(-5,0),(2,-1), \text { and }(4,3)$$
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