Problem 5 - Precalculus: Mathematical for Calculus

Question

Graph the inequality. $$y \leq 2 x+2$$

Step-by-Step Solution

Answer

Graph the line \(y = 2x + 2\) with a solid line and shade below it.

Step 1: Identify the Type of Line

Since the inequality is \(y \leq 2x + 2\), this indicates that you will graph a straight line. However, because it is "\( \leq \)", it means you will use a solid line to show that points on the line are also included in the solution.

Step 2: Find the Equation of the Line

To graph the inequality, start by graphing the corresponding equation without the inequality sign: \(y = 2x + 2\). This will help define the boundary line.

Step 3: Determine the Slope and y-Intercept

The equation is in slope-intercept form \(y = mx + b\) where \(m = 2\) is the slope and \(b = 2\) is the y-intercept. This means the line crosses the y-axis at (0,2) and rises 2 units for each 1 unit it moves to the right.

Step 4: Graph the Line

Start by plotting the y-intercept (0,2) on the graph. Then, use the slope to find another point by moving up 2 units and to the right 1 unit, plotting the second point at (1,4). Connect these points with a solid line.

Step 5: Shade the Appropriate Area

Since the inequality is \(y \leq 2x + 2\), you need to shade the area below the line. Choose a test point not on the line, like (0,0), substitute into the inequality, and verify: \(0 \leq 2(0) + 2\), which is true. Therefore, the region below the line is shaded.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of expressing linear equations using the formula \(y = mx + b\). In this formula:

• \(m\) is the slope, which tells you how steep the line is. It's the rate at which \(y\) changes as \(x\) changes.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

To graph the equation \(y=2x+2\), you start by identifying the y-intercept. Here, the y-intercept \(b = 2\), meaning the line crosses the y-axis at the point \((0, 2)\). Next, use the slope \(m = 2\), which means the line rises 2 units for every 1 unit it moves to the right. This slope direction helps in plotting another point, like moving from \((0, 2)\) to \((1, 4)\). Connect these points to form the line.

Linear Equations

Linear equations represent straight lines on a graph. They are called "linear" because they produce a straight line when graphed. Given in different forms, slope-intercept form is particularly useful for easily graphing these lines.Linear equations can describe relationships where one variable depends on another in a consistent way. They are used in many fields to solve practical problems, like predicting trends and modeling traffic.To graph a linear equation, form it as \(y = mx + b\). Recognize the characteristics within the equation:

• The slope \(m\) determines how the line angles.
• The y-intercept \(b\) helps you know where the line hits the y-axis.

Grasping these features lets you quickly draw and analyze the line's behavior and meaning.

Shading Regions

When dealing with inequalities, such as \(y \leq 2x + 2\), it's not just about graphing a line but also about understanding which part of the graph represents the solution. The inequality sign "\(\leq\)" tells you that you're dealing with a region, not just a line. Since it's "less than or equal to," you shade the area below the line on the graph. Choose a point outside the plotted line to check which area satisfies the inequality by substituting the point's coordinates into the inequality.If the inequality holds true at that point, it's part of the solution region. For instance, using the point \((0, 0)\):

• Plugging \((0, 0)\) into \(y \leq 2x + 2\) gives \(0 \leq 2(0) + 2\), which simplifies to \(0 \leq 2\) and is true.

This confirms your shading of the area below the line is accurate, visually representing where all the solutions to the inequality lie. Solid lines indicate that points on the line are solutions too, unlike dashed lines used for strict inequalities.