Question

Sketch the graph of the inequality. $$\frac{1}{3} x-\frac{2}{3} y \geq 2$$

Short answer

The first step involves rearranging the inequality to the form \(y \leq \frac{1}{2}x - 3\). Next, you plot the line using the y-intercept and slope. Since the inequality symbol is \(\leq\), the line is solid. Then, the area below the line is shaded to represent the solution to the inequality.

Step-by-step solution

Rearrangement to slope-intercept form

Firstly, rearrange the inequality \(\frac{1}{3} x - \frac{2}{3} y \geq 2\) to the slope-intercept form \(y = mx + c\). To do this, multiply every term by 3 which gives \(x - 2y \geq 6\). Further rearrange to isolate y, giving the equation as \(-2y \geq -x + 6\). Dividing every term by -2 to solve for \(y\), remember that when you divide by a negative number, the sign of inequality changes, so we get \(y \leq \frac{1}{2}x - 3\).

Sketch the line

Next, sketch the line \(y = \frac{1}{2}x - 3\). Begin by plotting the y-intercept which is -3. Then use the slope to find another point. The slope is \(\frac{1}{2}\), means that for every 2 units you move to the right (positive direction), you move up 1 unit. Hence, another point is (2, -2). Plot these points and draw the line. Since the inequality is \(\leq\), the line should be solid to indicate the points on the line satisfy the inequality.

Shade the area

The last step is to shade the area that represents the inequality \(y \leq \frac{1}{2}x - 3\). The inequality is saying that 'y' is less than or equal to something, this means that you will shade below the line.

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is crucial when graphing linear inequalities. It is the algebraic structure given as \(y = mx + c\), where \(m\) represents the slope of the line, and \(c\) the y-intercept, the point where the line crosses the y-axis.

To illustrate this, let's focus on an exercise example where you're required to sketch the graph of the inequality \(\frac{1}{3} x - \frac{2}{3} y \geq 2\). The first step here is to manipulate this inequality into the slope-intercept form. You multiply every term by 3 and rearrange the terms to isolate y, leading to \(y \leq \frac{1}{2}x - 3\). This now highlights a slope (\(m\)) of \(\frac{1}{2}\) and a y-intercept (\(c\)) of -3, making it simpler to plot on a graph.

Inequality Graphing

Graphing inequalities is similar to graphing linear equations, but it includes an additional step to represent the range of possible solutions. Unlike equations which have a single line, inequalities show a continuum of possible solutions, typically shaded on one side of the line.

In our example, after rearranging the inequality to slope-intercept form, the next task is sketching the line. You start by plotting the y-intercept at -3, then use the slope \(\frac{1}{2}\) to find another point. For every 2 units moved horizontally, you move 1 unit vertically. Our line is solid since the inequality includes equal to, meaning points on the line are part of the solution set. To represent the set of solutions graphically, you will need to shade the appropriate region.

Solving Inequalities

Solving inequalities means finding all its possible solutions. It requires isolating the variable of interest on one side of the inequality. Remember to switch the direction of the inequality sign whenever you multiply or divide by a negative number.

Using the training example, isolating y involved division by -2. Because we divided by a negative number, the inequality sign direction changed from \(\geq\) to \(\leq\). This reversal is a pivotal rule in solving inequalities and must always be observed to ensure correct solutions. Once isolated, you get a clear picture of how y is related to x in the inequality, paving the way for graphing.

Shading Inequality Regions

Shading is a way to visually depict all possible solutions to an inequality on a graph. The direction of the shading depends on the inequality symbol. If it’s '\(\geq\)' or '\(>\)', you shade above the graphed line; if it’s '\(\leq\)' or '\(<\)', you shade below.

In the case of our linear inequality \(y \leq \frac{1}{2}x - 3\), we shade below the line because the inequality specifies that y is less than, or equal to, a particular value. Therefore, any point in the shaded area, including those on the line due to the 'equal to' part of the inequality, would satisfy it. The shaded region tells us the combination of x and y values that are solutions to the inequality.