Problem 11 - Intermediate Algebra

Question

Graph each inequality. $$ y>x+1 $$

Step-by-Step Solution

Answer

Graph a dashed line for \( y = x+1 \) and shade above the line.

Step 1: Identify the Inequality Form

The given inequality is in the form of a linear inequality: \( y > x + 1 \). This can be related to the equation of a line \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Determine the Boundary Line

Convert the inequality into an equation: \( y = x + 1 \). This line is the boundary of our inequality. The inequality is strictly greater than, thus the line will be dashed, indicating that points on the line are not included in the solution set.

Step 3: Find the Boundary Line Points

Use the equation \( y = x + 1 \) to find two points on the line. For example, if \( x = 0 \), then \( y = 1 \), giving us the point \( (0, 1) \). If \( x = 1 \), then \( y = 2 \), giving us the point \( (1, 2) \).

Step 4: Graph the Boundary Line

Plot the points \( (0, 1) \) and \( (1, 2) \) on the coordinate plane. Draw a dashed line through these points to represent the boundary line \( y = x + 1 \).

Step 5: Choose a Test Point

Select a test point not on the line to determine which side of the line satisfies the inequality. A good test point is \( (0, 0) \), which is easy to substitute into the inequality.

Step 6: Test the Inequality

Substitute the test point \( (0, 0) \) into the inequality: \( 0 > 0 + 1 \). This simplifies to \( 0 > 1 \), which is false, indicating that the region containing \( (0, 0) \) is not part of the solution.

Step 7: Shade the Solution Region

Since the point \( (0, 0) \) does not satisfy the inequality, shade the region on the opposite side of the boundary line. This represents the solution where \( y > x + 1 \) is true.

Key Concepts

Linear equations

A linear equation forms the backbone of many mathematical concepts, especially when exploring inequalities. Essentially, a linear equation is a type of polynomial equation of the first degree. Such equations are typically expressed in the format: \[ y = mx + b \] where:

• \( y \) is the dependent variable.
• \( x \) is the independent variable.
• \( m \) represents the slope of the line.
• \( b \) signifies the y-intercept, indicating where the line crosses the y-axis.

To understand the role of linear equations in graphing inequalities, consider their application in delineating the boundary line. For example, the inequality \( y > x + 1 \) translates to the boundary line \( y = x + 1 \). Both the inequality and its related equation share the same slope and y-intercept. It is this relationship that allows us to efficiently graph an inequality by first plotting its counterpart linear equation.

Slope of a line

The slope of a line in a linear equation is represented by the variable \( m \). It provides valuable information about the direction and steepness of the line. In terms of definition, the slope is the ratio of the "rise" over the "run" between two distinct points on the line. This can be expressed through the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] where \((x_1, y_1)\) and \((x_2, y_2)\) are any two points on the line. The implication of the slope is straightforward:

• If \( m > 0 \), the line rises from left to right, indicating a positive slope.
• If \( m < 0 \), the line falls from left to right, implying a negative slope.
• When \( m = 0 \), the line is horizontal.
• A vertical line, which does not have a defined slope, corresponds to an undefined slope.

Using the inequality example \( y > x + 1 \), the slope \( m = 1 \) signifies the line rises diagonally across the coordinate plane. This understanding helps when graphing the boundary line for the associated inequality.

Coordinate plane

The coordinate plane is a two-dimensional space formed by two number lines: the x-axis and the y-axis. These axes intersect at the origin, marked as \((0, 0)\), and create four quadrants. These quadrants are valuable for plotting points and graphs:

• Quadrant I: where both \( x \) and \( y \) are positive.
• Quadrant II: where \( x \) is negative and \( y \) is positive.
• Quadrant III: where both \( x \) and \( y \) are negative.
• Quadrant IV: where \( x \) is positive and \( y \) is negative.

When graphing linear inequalities such as \( y > x + 1 \), you'll typically start by plotting a boundary line using the associated linear equation on this plane. Select distinct points (like \( (0, 1) \) and \( (1, 2) \)) and draw the line through them. For the inequality \( y > x + 1 \), the line should be dashed, signifying the boundary does not include the line itself. You then select a test point, like \( (0, 0) \), to determine which side of the line satisfies the inequality. Shade the appropriate side of the dashed line to represent the solution set. Understanding the coordinate plane is crucial for visualizing and solving such inequalities physically.