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Chapter 4: Problem 547

Graph the linear inequality \(y \leq \frac{5}{4} x\)

Short Answer

Expert verified
Plot the line \(y = \frac{5}{4} x\) solid and shade below it.

Step by step solution

01

Identify the Equation of the Boundary Line

The inequality given is in the form of a linear equation: \[y \leq \frac{5}{4} x\].First, convert it to the form of a line by considering the equation without the inequality: \[y = \frac{5}{4} x\].
02

Plot the Boundary Line

Graph the line \(y = \frac{5}{4}x\) on the coordinate plane. Since the inequality is \( \leq \), we will draw a solid line, indicating that points on the line itself satisfy the inequality.
03

Determine Which Side to Shade

To determine which side of the line should be shaded, choose a test point not on the boundary line, typically \((0,0)\) is an easy choice.Substitute \((0,0)\) into the inequality:\[0 \leq \frac{5}{4} \times 0\]\[0 \leq 0\]Since this is true, shade the region that includes the point \((0, 0)\).
04

Shade the Appropriate Region

Shade the region below and including the line \(y = \frac{5}{4} x\), since points in this region satisfy the inequality \(y \leq \frac{5}{4} x\).

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Key Concepts

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

Linear Equations
Linear equations are fundamental in algebra as they represent straight lines on a coordinate plane. A linear equation has the form: \( y = mx + b \). Here, \(m\) is the slope that indicates the steepness of the line, and \(b\) is the y-intercept where the line crosses the y-axis. In the given exercise, the equation is \( y = \frac{5}{4} x \), meaning the slope (m) is \(\frac{5}{4}\) and there is no y-intercept (b is 0). This tells us that the line passes through the origin (0, 0) and goes up 5 units for every 4 units it moves to the right.
Coordinate Plane
A coordinate plane is a two-dimensional plane formed by the intersection of a vertical line (y-axis) and a horizontal line (x-axis). It helps us visualize equations and their graphs. Each point on the plane is described by an ordered pair \((x, y)\). For example, the point \((3, 2)\) means moving 3 units to the right on the x-axis and 2 units up on the y-axis. In our exercise, plotting the boundary line \( y = \frac{5}{4} x \) would involve plotting multiple points like \((0,0)\), \((4, 5)\), and \((8, 10)\) and connecting them to form a straight line.
Inequality Shading
Inequality shading helps us show which part of the graph satisfies the inequality. The inequality in our exercise is \( y \leq \frac{5}{4} x \). After graphing the boundary line (the equation without the inequality), we need to determine which side of that line represents solutions to our inequality. We use a test point, like \( (0, 0) \), and plug it into the inequality: \( 0 \leq \frac{5}{4} \times 0 \), which is true. As a result, we shade the side containing \( (0, 0) \), which is below the line, indicating all those points satisfy the inequality.
Boundary Line
The boundary line is a crucial element when graphing inequalities. It's the line that you first graph without considering the inequality. In this exercise, the boundary line is \( y = \frac{5}{4} x \). The type of line (solid or dashed) matters: since our inequality signs include \( \leq \) or \( \geq \) (less than or equal to, or greater than or equal to), we use a solid line. It means points on the line satisfy the inequality. If the inequality had been strictly \( < \) or \( > \) (less than, or greater than), we鈥檇 use a dashed line. This indicates those points are not included in the solution.

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Most popular questions from this chapter

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