Problem 41 - Elementary Algebra

Question

$$ \text { For Problems 37-56, graph each linear inequality. (Objective } 3 \text { ) } $$ $$ 2 x-y \geq 4 $$

Step-by-Step Solution

Answer

Shade the area below the line \(2x - y = 4\); include points exactly on the line.

Step 1: Rewrite the Inequality as an Equation

First, rewrite the inequality as an equation by removing the inequality sign. This gives us: \(2x - y = 4\).

Step 2: Find the y-intercept and Slope

By rewriting the equation \(2x - y = 4\) in slope-intercept form \(y = mx + b\), we have: \(y = 2x - 4\). Here, the y-intercept \(b\) is \(-4\) and the slope \(m\) is \(2\).

Step 3: Plot the y-intercept

On a coordinate plane, plot the y-intercept \((0, -4)\) as your starting point.

Step 4: Use the Slope to Find Another Point

From the y-intercept \((0, -4)\), use the slope \(2\), which means "rise 2, run 1," to find another point. Starting from \((0, -4)\), move up 2 units and right 1 unit to reach \((1, -2)\). Plot this point.

Step 5: Draw the Line

Draw a solid line through the points \((0, -4)\) and \((1, -2)\). The solid line indicates the 'equals to' part of the inequality \( \geq \).

Step 6: Decide Which Side to Shade

To determine which side of the line to shade, choose a test point not on the line, such as \((0, 0)\). Substitute into the inequality: \(2(0) - 0 \geq 4\) simplifies to \(0 \geq 4\), which is false. Thus, the region opposite the test point (below the line) is shaded.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a convenient way to express lines using the formula \(y = mx + b\). This format allows us to easily identify two crucial pieces of the line's properties: the slope \(m\) and the y-intercept \(b\).
The slope \(m\) represents the steepness of the line or how much the line rises or falls as it moves horizontally. If \(m\) is positive, the line ascends from left to right; if negative, it descends.

• **Slope**: Tells us how sharply the line increases or decreases.
• **Y-intercept** \(b\): The point where the line crosses the y-axis, telling us the vertical position of the line when \(x = 0\).

To convert an equation into slope-intercept form, rearrange it to solve for \(y\). This format simplifies the graphing process.

Coordinate Plane

The coordinate plane forms the foundation of graphing lines and inequalities. It consists of two perpendicular lines called axes: the horizontal x-axis and the vertical y-axis. These axes intersect at the origin, denoted by \((0, 0)\).
Each point on the plane corresponds to an \((x, y)\) coordinate, where \(x\) specifies the horizontal position and \(y\) indicates the vertical position. The plane is divided into four quadrants:

• **Quadrant I**: Located in the top-right, where both \(x\) and \(y\) are positive.
• **Quadrant II**: Top-left, with \(x\) negative and \(y\) positive.
• **Quadrant III**: Bottom-left, both \(x\) and \(y\) are negative.
• **Quadrant IV**: Bottom-right, \(x\) is positive and \(y\) is negative.

Understanding how points and lines are located on the coordinate plane is essential for graphing linear equations and inequalities.

Plotting Points

Plotting points on the coordinate plane allows us to visually represent mathematical relationships. To plot a point, start at the origin \((0, 0)\) and move along the x-axis by the \(x\)-coordinate value. Then, shift vertically by the \(y\)-coordinate value< Fibonacci sequence?">:
For example, plotting the point \((1, -2)\) involves moving 1 unit to the right on the x-axis, then 2 units down since the y-coordinate is negative. Each point is represented by a dot at the corresponding \((x, y)\) location.
Plotting helps establish reference points, which aid in drawing lines and interpreting relationships between variables in a graph.

Shading Regions in Inequalities

Shading regions when graphing inequalities visually represents the range of solutions. For the inequality \(2x - y \geq 4\), the solution includes all points that satisfy the condition. After plotting points to form the line, we determine which side of the line contains valid solutions.
To shade effectively, use a test point not on the line, like \((0, 0)\). Substitute these coordinates into the inequality:

• For our example \(2(0) - 0 \geq 4\), which simplifies to \(0 \geq 4\) (false), the shading is opposite the test point.

The chosen side that fails the test indicates that the other side contains solutions to the inequality. Shade that region to communicate where all solutions lie, visually indicating whether an inequality's solution set is above or below a particular line.