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Chapter 5: Problem 28

Graph the solution set. \(y \leq 5\)

Short Answer

Expert verified
Draw a solid horizontal line at \(y = 5\) and shade below it.

Step by step solution

01

Understand the Inequality

The inequality given is \(y \leq 5\). This inequality means that the values of \(y\) should be less than or equal to 5.
02

Draw the Boundary Line

Draw a horizontal line at \(y = 5\) on the graph. Since the inequality is \(\leq\), this line will be solid, indicating that points on the line are included in the solution set.
03

Shade the Solution Region

Shade the entire region below the horizontal line \(y = 5\). This shading represents all points where \(y\) is less than or equal to 5.

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Key Concepts

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

Inequalities
When working with inequalities, it's important to understand what they represent. An inequality like \(y \leq 5\) indicates that \(y\) can be any number that is either less than or equal to 5. This is different from an equation, where \(y = 5\) would limit \(y\) to just one value.

Inequalities can use different symbols:
  • \(< \) to mean 'less than'
  • \(>\) to mean 'greater than'
  • \(\leq\) to mean 'less than or equal to'
  • \(\geq\) to mean 'greater than or equal to'
Understanding these symbols helps us correctly represent and interpret the solutions.
Graphing
Graphing inequalities involves a few clear steps. To graph \(y \leq 5\), first draw the boundary line that the inequality describes. In this case, it is the horizontal line \(y = 5\). This line is solid because the inequality is \(\leq\), meaning points on the line are included.

After drawing the boundary line, the next step is to shade the correct region. Since \(y\) can be less than or equal to 5, we shade below the line. This shading visually represents all the possible values of \(y\) that satisfy the inequality.
Solution Set
The solution set is the collection of all points that satisfy the inequality \(y \leq 5\). On the graph, the solution set includes:
  • All points on the boundary line \(y = 5\)
  • All points below this boundary line
Visually, the shaded region and the line together show all possible solutions.

Remember, for a different inequality such as \(y < 5\), the boundary line would be dashed, indicating that points on the line \(y = 5\) are not included in the solution set. Therefore, shading and line types are crucial in conveying the exact nature of the inequality.

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Most popular questions from this chapter

Michelle borrows a total of \(\$ 5000\) in student loans from two lenders. One charges \(4.6 \%\) simple interest and the other charges \(6.2 \%\) simple interest. She is not required to pay off the principal or interest for 3 yr. However, at the end of 3 yr, she will owe a total of \(\$ 762\) for the interest from both loans. How much did she borrow from each lender?

Two particles begin at the same point and move at different speeds along a circular path of circumference \(280 \mathrm{ft}\). Moving in opposite directions, they pass in \(10 \mathrm{sec} .\) Moving in the same direction, they pass in \(70 \mathrm{sec} .\) Find the speed of each particle.

Solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples \(5-6\) ) $$ \begin{array}{l} 2 x+4=4-5 y \\ 2+4(x+y)=7 y+2 \end{array} $$

Solve the system using any method. $$ \begin{array}{l} \frac{1}{14} x-\frac{1}{7} y=\frac{1}{2} \\ 2(x-2 y)+3=20 \end{array} $$

Solve the system using any method. $$ \begin{array}{l} \frac{x-2}{8}+\frac{y+1}{2}=-6 \\ \frac{x-2}{2}-\frac{y+1}{4}=12 \end{array} $$
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