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Chapter 4: Problem 33

Graph the inequality \(y \geq-x^{2}.\)

Short Answer

Expert verified
Shade the region above and on the parabola \( y = -x^2 \)

Step by step solution

01

Identify the equation for the boundary

The given inequality is in the form of a quadratic inequality. The boundary of this inequality is represented by the equation: \[ y = -x^2 \]
02

Graph the boundary parabola

Graph the equation \( y = -x^2 \). This parabola opens downwards (since the coefficient of \( x^2 \) is negative) and passes through the origin (0,0). Plot a series of points, such as \((-2, -4)\), \((-1, -1)\), \((0, 0)\), \((1, -1)\), and \((2, -4)\), and draw the parabola through these points.
03

Determine the shading region

The inequality \( y \geq -x^2 \) indicates that the region above or on the parabola should be shaded. This includes all the points on the parabola and those above it.
04

Verify with a test point

Pick a test point not on the boundary, such as \((0,1)\). Substitute into the inequality: \[ 1 \geq -(0)^2 \] This simplifies to: \[ 1 \geq 0 \] which is true. Hence, the region including \((0,1)\) is part of the solution.

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Key Concepts

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

Quadratic Inequality
A quadratic inequality is an inequality that involves a quadratic expression. It typically has a form like y greater or equal - x squared.

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Most popular questions from this chapter

Recall that the absolute value of a number is its distance from 0 on the number line. You can solve equations involving absolute values. For example, the solutions of the equation \(|x|=8\) are the two numbers that are a distance of 8 from 0 on the number line, 8 and \(-8 .\) Solve each equation. $$ \begin{array}{ll}{\text { a. }|a|=2.5} & {\text { b. }|2 b+3|=8} \\ {\text { c. }|9-3 c|=6} & {\text { d. } \frac{15 d 1}{25}=1} \\ {\text { e. }|-3 e|=15} & {\text { f. } 20+|2.5 f|=80}\end{array} $$

Examine this table. a. Use the table to estimate the solutions of \(t(t-3)=5\) to the nearest integer. b. If you were searching for solutions by making a table with a calculator, what would you have to do to find solutions to the nearest tenth? c. Find two solutions of \(t(t-3)=5\) to the nearest tenth. $$ \begin{array}{rr}{t} & {t(t-3)} \\ {-2} & {10} \\ {-1} & {4} \\ {0} & {0} \\\ {1} & {-2} \\ {2} & {-2} \\ {3} & {0} \\ {4} & {4} \\ {5} & {10} \\\ {6} & {18} \\ {7} & {28}\end{array} $$

A group of friends enters a restaurant. No table is large enough to seat the entire group, so the friends agree to sit at several separate tables. They want to sit in groups of \(5,\) but there aren't enough tables: 4 people wouldn't have a place to sit. Someone suggests they sit in groups of \(6,\) which would fill all the tables, with 2 extra seats at one table. Answer these questions to find how many people are in the group and how many tables are available. a. Write a system of two equations to describe the situation. b. Solve your system of equations. How many friends are in the group, and how many tables are available? Check your work.

Solve each equation by doing the same thing to both sides. $$ 7 y-4=4 y-13 $$

Sort these expressions into two groups so that the expressions in each group are equal to one another. $$ m^{3} \quad\left(\frac{1}{m}\right)^{3} \quad m^{-3} \quad\left(\frac{1}{m}\right)^{-3} \quad \frac{1}{m^{3}} \quad m \div m^{4} $$
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