Problem 13 - Intermediate Algebra

Question

Graph each inequality. $$ y \geq-\frac{3}{2} x+1 $$

Step-by-Step Solution

Answer

Shade the region above and including the line through (0,1) and (2,-2) on the graph.

Step 1: Identify the Inequality Type

This inequality involves a linear equation, where the inequality symbol is 4. This means that the line will be solid (indicating that points on the line satisfy the inequality) and the area we shade will be above the line.

Step 2: Convert into Slope-Intercept Form

The inequality is already in slope-intercept form, which is given as: \[ y \geq -\frac{3}{2}x + 1 \]Here, the slope \( m = -\frac{3}{2} \) and the y-intercept \( b = 1 \).

Step 3: Find the Y-Intercept

The y-intercept of the line is where the line crosses the y-axis, which occurs when \( x = 0 \). Using the equation:\[ y = -\frac{3}{2}(0) + 1 = 1 \]This means the y-intercept is the point (0, 1).

Step 4: Plot the Y-Intercept

Start by plotting the y-intercept point (0,1) on the graph.

Step 5: Use the Slope to Find Another Point

Slope \( m = -\frac{3}{2} \) can be interpreted as "down 3 units for every 2 units to the right." Starting from the y-intercept (0,1), move 2 units to the right (to x=2) and then 3 units down to find the corresponding point (2,-2).

Step 6: Plot the Second Point

Plot the second point (2,-2) that you found using the slope on the graph.

Step 7: Draw the Line

Draw a solid line through the points (0,1) and (2,-2). The line should extend to cover the entire graph area and indicate that the line is part of the solution set due to the \( \geq \) symbol.

Step 8: Shade the Solution Area

Since the inequality is \( y \geq -\frac{3}{2}x + 1 \), shade the region above and including the line. This area represents all the points \((x,y)\) that satisfy the inequality.

Key Concepts

Linear Equations

Linear equations form the backbone of graphing inequalities. They come in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Linear equations are called "linear" because they graph as straight lines on the coordinate plane. They depict a constant rate of change, represented by the slope. A linear equation does not curve or create any bends, hence its name.Graphing a linear equation involves plotting points that satisfy the equation and drawing a line through these points. For many students, understanding linear equations is key before delving into inequalities. This is because inequalities use the principles of linear equations as their foundation but add additional complexity with the inequality symbol.

Slope-Intercept Form

The slope-intercept form is exceptionally useful for graphing linear equations and inequalities. It is represented by the formula \( y = mx + b \). Here, \( y \) and \( x \) are variables, \( m \) is the slope, and \( b \) is the y-intercept.The slope \( m \) indicates how steep the line is, and whether it rises or falls as you move along the x-axis:

• Positive slope: Line ascends from left to right.
• Negative slope: Line descends from left to right.
• Slope of zero: Line is horizontal.

The y-intercept \( b \) shows where the line intersects with the y-axis. This form makes it simple to determine these key features directly from the equation.For graphing inequalities, the slope-intercept form is used for both drawing the boundary line and identifying where to begin shading.

Y-Intercept

The y-intercept of a linear equation or inequality is a significant starting point. It is denoted by the "\( b \)" in the slope-intercept form \( y = mx + b \). The y-intercept is the point where the line crosses the y-axis, where the x-value is 0.Finding the y-intercept involves evaluating the equation when \( x = 0 \). For example, if \( y = -\frac{3}{2}x + 1 \), the y-intercept would be at the point \( (0, 1) \). This is because when you insert \( x = 0 \) into the equation, \( y = 1 \).It is crucial to mark the y-intercept on the graph as it becomes the base point from which the line is drawn using the slope. It acts as a reference for finding other points on the line.

Shading Region in Graph

Shading is a critical step when graphing inequalities. It visually represents the set of solutions that satisfy the inequality. Once you have graphed the line based on the equation \( y = mx + b \), you use the inequality symbol to decide which side of the line to shade.For the inequality \( y \geq -\frac{3}{2}x + 1 \), the "\( \geq \)" symbol indicates you should shade above and on the line. This is because it includes all points where \( y \) is greater than or equal to \( -\frac{3}{2}x + 1 \).Here are some tips for shading:

• Use a solid line if the inequality includes \( \geq \) or \( \leq \), which means the line is part of the solution.
• Shade above the line for \( y \geq \) inequalities and below for \( y \leq \) inequalities.
• If stuck, use a test point not on the line, such as \( (0,0) \), to see if the inequality holds true and decide where to shade.

Shading effectively displays the entire solution set for an inequality, making it easier to understand what values fulfill the inequality.