Question

Sketch the graph of the inequality in a coordinate plane. $$ \frac{3}{4} x+\frac{1}{4} y \geq 1 $$

Short answer

The boundary line is a solid line starting from the y-intercept 4 and descends at a slope of -3. The area above the line (not containing the origin) is shaded indicating all the possible solution points for the inequality are in the shaded region.

Step-by-step solution

Convert the inequality into a linear equation

The first need is to isolate y. Dividing the equation by 1/4, you get: \( y \geq -3x + 4\).

Identify the slope and the y-intercept

In the equation, y ≥ -3x + 4, the coefficient of x is -3 (the slope, m) and the constant term is 4 (the y-intercept, b).

Sketch the boundary line

Now, you sketch a line with a slope of -3 and y-intercept of 4. Since the inequality contains a ≥ symbol, the line should be solid, indicating that points on the line are part of the solution.

Identify which side of the line to shade

For determining to shade which side of the line, use a test point. The origin (0,0) is often the easiest point to use, provided that it is not on the line. If we place x=0 and y=0 into the inequality, we get 0 ≥ 1, which is false. So, you have to shade the side of the line that does not contain the origin.

Key concepts

Coordinate Plane

Understanding the coordinate plane is essential for graphing and analyzing linear inequalities and equations. Imagine a flat surface where every point is defined by an X (horizontal) and a Y (vertical) coordinate. This surface is divided into four quadrants by two crossing lines: the x-axis (horizontal line) and the y-axis (vertical line). Points on the right of the y-axis have positive x coordinates, while points on the left have negative x coordinates. Similarly, points above the x-axis have positive y coordinates, and those below carry negative y coordinates.

When we graph the equation of an inequality, like \(\frac{3}{4} x + \frac{1}{4} y \geq 1\), we start by translating the algebraic expression into a visual representation on this plane. To accurately sketch the graph of the inequality, it is crucial to identify certain elements, such as the slope of the line and its y-intercept, which will guide us to draw the line across the plane. Once the line is drawn, we determine which side of the line represents the solution to the inequality by shading it accordingly.

Slope of a Line

The slope of a line indicates how steep the line is and its orientation on the coordinate plane. It is often denoted as 'm' and is calculated as the change in the Y coordinates over the change in the X coordinates between any two points on the line. In other words, it's the ratio of the vertical 'rise' to the horizontal 'run' between two points on the line.

In our exercise, given the inequality \( y \geq -3x + 4 \), the coefficient of 'x' is -3, which is the slope of the line. This number tells us that for every unit we move to the right (along the positive X-axis), we move down 3 units (along the negative Y-axis) since the slope is negative. If the slope were positive, the line would rise as we moved to the right. The concept of slope is pivotal because it defines the angle and direction of the line on the coordinate plane, which is crucial when graphing a linear equation or inequality.

Y-Intercept

The y-intercept is the point where the line crosses the y-axis on the coordinate plane. It’s where the line 'intercepts' or intersects the Y-axis, and consequently, its X-coordinate is always zero. The y-intercept is useful because it gives us a fixed point from which to begin drawing our line. In the context of the equation \( y = -3x + 4 \), the y-intercept is the constant term 4. This means that the point (0, 4) lies on the line.

When graphing inequalities like \(\frac{3}{4} x + \frac{1}{4} y \geq 1\), we start by plotting the y-intercept. It serves as an anchor for our line, ensuring it fits the equation or inequality properly. Moreover, the y-intercept provides a quick and clear indication of where the line will cross the Y-axis, providing a starting point for graphing the rest of the line using the slope. Understanding the importance of the y-intercept allows students to confidently approach the graphing of linear equations and inequalities.