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Chapter 5: Problem 9

For a constant real number \(k\), the inequality \(y>k\) represents the half-plane (above/below) the (horizontal/ vertical) line \(y=k\).

Short Answer

Expert verified
Above the horizontal line.

Step by step solution

01

Identify the Inequality

The given inequality is \(y > k\). Inequalities show one side of a boundary line where the inequality holds true.
02

Determine the Boundary Line

The boundary line for the inequality \(y > k\) is the line \(y = k\). This is a horizontal line.
03

Identify the Half-Plane

For \(y > k\), the region that satisfies the inequality lies above the horizontal line \(y = k\), because as \(y\) increases, it becomes greater than \(k\).

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Key Concepts

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

boundary lines
Inequalities in algebra often involve a boundary line. This line represents where the two sides of the inequality are equal. For instance, in the inequality \(y > k\), the boundary line is the line \(y = k\). Think of this line like a fence that divides a graph into two distinct regions. On one side of the boundary line, the inequality \(y > k\) holds true, and on the other side, it does not.
half-plane regions
When graphing inequalities, we use the boundary line to divide the coordinate plane into two half-plane regions. One region satisfies the inequality, and the other does not. For example, with \(y > k\), the half-plane that satisfies the inequality is the region above the boundary line \(y = k\). If you pick any point in this region, it will have a \(y\)-value greater than \(k\). To make sure you shade the correct region, substitute a point that is not on the boundary line back into the inequality. If the inequality holds true, shade that side of the line.
horizontal lines
A horizontal line is a straight line that runs left to right and is parallel to the x-axis. In the inequality \(y > k\), the boundary line \(y = k\) is a horizontal line. This means that every point on the line has the same \(y\)-value (which is \(k\)), no matter the \(x\)-value. Horizontal lines are important when graphing inequalities because they help us easily identify which half-plane region to shade. Remember, for \(y > k\), you shade the area above the horizontal line \(y = k\), whereas for \(y < k\), you would shade below.

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Most popular questions from this chapter

A system of equations is given in which each equation is written in slope- intercept form. Determine the number of solutions. If the system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. $$ \begin{array}{l} y=\frac{2}{5} x-7 \\ y=\frac{1}{4} x+7 \end{array} $$

Use the given constraints to find the maximum value of the objective function and the ordered pair \((x, y)\) that produces the maximum value. \(x \geq 0, y \geq 0\) \(3 x+4 y \leq 48\) \(2 x+y \leq 22\) \(y \leq 9\) a. Maximize: \(z=100 x+120 y\) b. Maximize: \(z=100 x+140 y\)

Determine if the ordered pair is a solution to the system of equations. (See Example 1\()\) \(y=-\frac{1}{5} x+2\) \(2 x+10 y=10\) a. (5,1) b. (-10,4)

Solve the system. $$ \begin{array}{l} 3^{x}-9^{y}=18 \\ 3^{x}+3^{y}=30 \end{array} $$

Josh makes \(\$ 24 /\) hr tutoring chemistry and \(\$ 20 / \mathrm{hr}\) tutoring math. Let \(x\) represent the number of hours per week he spends tutoring chemistry. Let \(y\) represent the number of hours per week he spends tutoring math. a. Write an objective function representing his weekly income for tutoring \(x\) hours of chemistry and \(y\) hours of math. b. The time that Josh devotes to tutoring is limited by the following constraints. Write a system of inequalities representing the constraints. \- The number of hours spent tutoring each subject cannot be negative. \- Due to the academic demands of his own classes he tutors at most \(18 \mathrm{hr}\) per week. \- The tutoring center requires that he tutors math at least 4 hr per week. \- The demand for math tutors is greater than the demand for chemistry tutors. Therefore, the number of hours he spends tutoring math must be at least twice the number of hours he spends tutoring chemistry. c. Graph the system of inequalities represented by the constraints. d. Find the vertices of the feasible region. e. Test the objective function at each vertex. f. How many hours tutoring math and how many hours tutoring chemistry should Josh work to maximize his income? g. What is the maximum income? h. Explain why Josh's maximum income is found at a point on the line \(x+y=18\).
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