Problem 7 - Elementary Algebra

Question

Solve the inequalities by graphing. $$ -x+2 y+4 \geq 0 $$

Step-by-Step Solution

Answer

Based on the given inequality to solve using a graph: Original inequality: $-x + 2y + 4 \geq 0$ 1. Rewrite the inequality in slope-intercept form: $y \geq \frac{1}{2}x - 2$ 2. Graph the corresponding linear equation: $y = \frac{1}{2}x - 2$ 3. Determine the type of line (solid or dashed) to use: Solid line 4. Shade the correct region and find the solution: Region above the solid line, including the line itself Using a graph, the solution to the inequality is the region above the solid line with the equation $y = \frac{1}{2}x - 2$, including the points on the line.

Step 1: Rewrite the inequality in slope-intercept form

To rewrite the inequality in slope-intercept form, we want to isolate y on one side. The inequality is currently given as: $$ -x + 2y + 4 \geq 0 $$ Now, let's move the x-term and the constant term to the other side of the inequality and then divide by 2 to isolate y: $$ 2y \geq x - 4 $$ $$ y \geq \frac{1}{2}x - 2 $$ Now the inequality is in the slope-intercept form.

Step 2: Graph the corresponding linear equation

Replace the inequality sign with an equal sign to get the corresponding linear equation: $$ y = \frac{1}{2}x - 2 $$ This equation represents a line with a slope (m) of 1/2 and y-intercept (b) of -2. To graph this line, start at the y-intercept (-2) and use the slope to find more points on the line. Since the slope is 1/2, for each 1 unit increase in x, y will increase by 1/2.

Step 3: Determine the type of line (solid or dashed) to use

Since the inequality in the original problem has a "greater than or equal to" sign (≥), we will use a solid line on the graph to represent that points on the line are included in the solution set.

Step 4: Shade the correct region and find the solution

To shade the correct region, use a test point that does not lie on the line (such as the origin, (0, 0)) and substitute the x and y values into the inequality. If the point satisfies the inequality, shade the region that contains the test point. If it does not satisfy the inequality, shade the opposite region. Test point: (0, 0) $$ y \geq \frac{1}{2}x - 2 $$ $$ 0 \geq \frac{1}{2}(0) - 2 $$ $$ 0 \geq -2 $$ Since this is true, shade the region containing the origin. The solution to the inequality is the region above the solid line and includes the line itself.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to express linear equations, making it easy to visualize and graph them. It is given by the formula \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept.
This form is particularly useful when solving inequalities because it allows us to quickly identify how the line will look on a graph.

• **Slope \(m\):** This tells us how steep the line is. A larger absolute value means a steeper line.
• **Y-intercept \(b\):** This is where the line crosses the y-axis.

Rewriting an equation or inequality in this form is the first step in graphing because it tells us exactly where to start and gives a clear direction of how the line should extend.

Linear Equations

Linear equations represent straight lines on a graph, and they form the backbone of many mathematical concepts. When you see an equation like \(y = \frac{1}{2}x - 2\), you're looking at a linear relationship. The key features include:

• **Straight Line:** Regardless of the specific numbers, the graph of a linear equation is always a straight line.
• **Constant Rate of Change:** The slope, \(m\), indicates that the rate at which \(y\) changes with respect to \(x\) is constant.

Understanding linear equations is crucial before diving into graphing inequalities, as inequalities build upon these same principles but add the dimension of direction and region.

Graphing Inequalities

Graphing inequalities involves a few extra steps beyond simply graphing a line. Once you have your line in slope-intercept form, here’s what to remember:

• **Solid vs. Dashed Line:** The type of line represents whether the points on the line are included in the solution set. A solid line, like in our example with \(y \geq \frac{1}{2}x - 2\), indicates that the line itself is part of the solution.
• **Shading the Region:** After drawing the line, the next step is to shade the appropriate side of it. This shaded area represents all the possible solutions to the inequality. A test point, often \((0, 0)\), can help determine which side to shade. If the test point satisfies the inequality, shade the side containing it.

This extra layer of shading transforms this process from graphing a single line to finding a region of solutions, making inequalities unique.

Solution Set

The solution set of an inequality includes all the possible \((x, y)\) pairs that satisfy the inequality. In the context of graphing, this manifests as a shaded region on the graph.
The line itself can belong to the solution set if the inequality is inclusive (\(\geq\) or \(\leq\)). In our example, the solution set is the area above the line \(y = \frac{1}{2}x - 2\) and includes all the points on the line itself.
By using a graph, we visually capture all solutions at once, which is highly efficient compared to calculating each solution point-by-point. This graphical representation is particularly useful for complex systems of inequalities, where colored regions clearly depict the areas of overlap and intersection.