Problem 64 - Algebra 2

Question

Graph each inequality. $$ y \leq-4 x+5 $$

Step-by-Step Solution

Answer

Graph the line \( y = -4x + 5 \) solid and shade below.

Step 1: Understand the Inequality

The given inequality is \( y \leq -4x + 5 \). This implies that we need to graph the line \( y = -4x + 5 \) and then shade the region where \( y \) is less than or equal to this expression.

Step 2: Identify the Boundary Line

The boundary line can be derived from the equation \( y = -4x + 5 \). This will be a straight line with a slope of \(-4\) and a y-intercept of \(5\). The line will be solid, not dashed, because the inequality uses \( \leq \).

Step 3: Plot the Y-Intercept

The y-intercept is the point where the line crosses the y-axis, which is \(5\). Plot this point on your graph at \( (0, 5) \).

Step 4: Use the Slope to Find Another Point

The slope of \(-4\) means that for every step right on the x-axis, you move 4 steps down on the y-axis. Starting from the y-intercept, move 1 unit to the right (x-axis) and 4 units down (y-axis) to find another point on the line. Plot this second point, \( (1, 1) \).

Step 5: Draw the Boundary Line

Connect the points \( (0, 5) \) and \( (1, 1) \) with a solid straight line. This reflects all points that satisfy the equation \( y = -4x + 5 \).

Step 6: Shade the Inequality Region

Since the inequality is \( y \leq -4x + 5 \), you need to shade the region below the line, including the line itself. This represents all the points that satisfy the inequality.

Key Concepts

Linear Equations

A linear equation represents a straight line on a graph. It is typically expressed in the form \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. Linear equations have constant slopes and do not curve on a graph. When you have two variables, such as \( y \) and \( x \), their relationship can be mapped as a straight line, illustrating a constant rate of change.
The equation \( y = -4x + 5 \) is a linear equation because it graphs as a straight line. In this case, it defines a direct correlation between the values of \( x \) and \( y \).

• "\( y \)" represents the dependent variable, changing in response to \( x \).
• "\( x \)" represents the independent variable.
• The graph's shape is determined by the coefficients and the constant term.

Understanding linear equations is crucial as it lays the groundwork for graphing inequalities.

Slope-Intercept Form

The slope-intercept form is one of the most common ways to write the equation of a line. It is expressed as \( y = mx + b \), where:

• "\( m \)" represents the slope of the line.
• "\( b \)" represents the y-intercept, which is the point where the line crosses the y-axis.

The slope \( m \) describes how steep the line is. A larger absolute value indicates a steeper line. In the inequality \( y \leq -4x + 5 \), the slope is \(-4\), indicating a downward trend, as you would move 4 units down for every unit you move right.
The y-intercept \( 5 \) tells us where the line crosses the y-axis. In practical terms, it's the value of \( y \) when \( x = 0 \). This form provides a simple way to quickly graph a linear equation by starting at "\( b \)" on the y-axis and using the slope \( m \) to determine the line's direction.

Boundary Lines

In graphing inequalities, boundary lines are used to separate regions on a graph. The boundary line represents the equation part of the inequality (like \( y = -4x + 5 \) in our example), showing the dividing line between where the inequality holds true and false.
The type of boundary line depends on the inequality sign:

• "\( < \)" or "\( > \)" will have dashed lines, indicating the line itself is not included in the solution.
• "\( \leq \)" or "\( \geq \)" will have solid lines, indicating the line is part of the solution set.

In the inequality \( y \leq -4x + 5 \), the boundary line is solid because \( y \) can equal \(-4x + 5\). This means the line itself is part of the solution, and the area of interest includes this line and the area below it.

Shading Regions

Shading regions on a graph indicates which side of the boundary line holds the solution to the inequality. Once you have plotted the line, you need to decide where to shade to show all the values satisfying the inequality.
The direction of shading depends on the inequality sign:

• "\( > \)" or "\( \geq \)": Shade above the line.
• "\( < \)" or "\( \leq \)": Shade below the line.

For \( y \leq -4x + 5 \), after drawing the boundary line, the correct region to shade is below the line. This shaded section on the graph represents all potential solutions where \( y \) is less than or equal to \(-4x + 5\).
Shading clearly communicates which part of the graph aligns with the inequality, making it easier to understand and verify solutions.