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Chapter 5: Problem 90

What does a solid line mean in the graph of an inequality?

Short Answer

Expert verified
In the graph of an inequality, a solid line indicates that the points on the line are included in the solution of the inequality. It is usually used for 'greater than or equal to' or 'less than or equal to' inequalities.

Step by step solution

01

Understanding Inequalities and Their Graphs

An inequality compares two values, showing if one is less than, greater than, or simply not equal to another value. An inequality can be represented on a number line, in which the numbers that make the inequality true are shaded or circled. Inequalities can also be represented on a xy-plane, often used when you have a two-variable inequality.
02

Explaining Solid Lines

When graphing inequalities on a xy-plane, a solid line is used to represent a part of the graph where the points on the line are included in the solution of the inequality. This is usually the case for 'greater than or equal to' (\(>=\)) or 'less than or equal to' (\(<=\)) inequalities.
03

Differentiating from Dashed Lines

In contrast, a dashed line is used when the points on the line are not included in the solution, applicable for 'greater than' (\(>\)) or 'less than' (\(<\)) inequalities. This differentiation is crucial to accurately understanding and interpreting graphed inequalities.

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Most popular questions from this chapter

In Exercises \(43-46,\) let \(x\) represent one number and let \(y\) represent the other number. Use the given conditions to write a system of equations. Solve the system and find the numbers. The sum of two numbers is \(7 .\) If one number is subtracted from the other, their difference is \(-1 .\) Find the numbers.

You throw a ball straight up from a rooftop. The ball misses the rooftop on its way down and eventually strikes the ground. A mathematical model can be used to describe the relationship for the ball's height above the ground, \(y,\) after \(x\) seconds. Consider the following data: $$\begin{array}{cc} \hline x, \text { seconds after the ball is } & y, \text { ball's height, in feet, above } \\ \text { thrown } & \text { the ground } \\ \hline 1 & 224 \\ 3 & 176 \\ 4 & 104 \end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=5 .\) Describe what this means.

A mathematical model can be used to describe the relationship between the number of feet a car travels once the brakes are applied, \(y,\) and the number of seconds the car is in motion after the brakes are applied, \(x .\) A research firm collects the following data: $$\begin{array}{cc} \hline \begin{array}{c} x \text { , seconds in motion } \\ \text { after brakes are applied } \end{array} & \begin{array}{c} y, \text { feet car travels } \\ \text { once the brakes are applied } \end{array} \\ \hline 1 & 46 \\ 2 & 84 \\ 3 & 114 \end{array}$$ a. Find the quadratic function \(y=a x^{2}+b x+c\) whose graph passes through the given points. b. Use the function in part (a) to find the value for \(y\) when \(x=6 .\) Describe what this means.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I need to be able to graph systems of linear inequalities in order to solve linear programming problems.

Graph each inequality. $$y
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