Problem 17 - Elementary Algebra

Question

Graph the linear equations and inequalities. $$ y-5<3 $$

Step-by-Step Solution

Answer

Answer: The graph of the linear inequality \(y-5<3\) consists of a dashed horizontal line at \(y=8\), and the solution region is the area below this line, representing all points with y-values less than 8.

Step 1: Rewrite the inequality in the slope-intercept form

We need to rewrite inequality \(y-5<3\) in the slope-intercept form (\(y=mx+b\)). To do this, we need to isolate the y. So in this case, we have: $$ y<3+5 $$ $$ y<8 $$ Now, the inequality is in the slope-intercept form (\(y=8\)).

Step 2: Sketch the boundary line

Next, we will draw the boundary line \(y=8\). Since the inequality is less than (\(<\)) and not less than or equal to (\(\le\)), we will draw a dashed (not solid) line at \(y=8\).

Step 3: Shade the solution region

Now we need to shade the solution region for the inequality. Since the inequality states that \(y<8\), we will shade the region below the dashed line at \(y=8\). This shaded region represents all the points where the y values are less than 8 and satisfy the given inequality. That's it! We have graphed the linear inequality \(y-5<3\). The dashed line at \(y=8\) represents the boundary, and the shaded region below the line represents the solution to the inequality.

Key Concepts

Graphing Inequalities

Graphing inequalities is a crucial skill in algebra. When we graph an inequality, we effectively show all the possible solutions that satisfy the inequality on a coordinate plane. For instance, in the case of the inequality \(y < 8\), we are interested in all the values of \(y\) that are less than 8. When graphing inequalities, you need to keep in mind a few key steps:

• First, graph the related equation as if the inequality were an equality. For \(y < 8\), this related equation is \(y = 8\).
• If the inequality is strict, such as \(>\) or \(<\), use a dashed line to indicate that points on this line are not included in the solution set. If the inequality includes equal to, such as \(\leq \) or \(\geq \), use a solid line to include all points on the line in the solution set.
• Shade the area of the graph where the inequality holds true. This shaded region represents all possible solutions. Keep a watch on the inequality sign and shade the correct side of the boundary line.

By correctly applying these steps, you can visually interpret and solve linear inequalities.

Slope-Intercept Form

The slope-intercept form is a standard way of writing the equation of a line. It makes graphing the line very straightforward. The general format is \(y = mx + b\), where \(m\) represents the slope of the line, and \(b\) is the y-intercept. For our inequality \(y < 8\), the slope-intercept form is particularly simple because there is no slope. However, understanding this form helps as:

• The y-intercept \(b=8\) directly tells us where the line intersects the y-axis. In our equation \(y=8\), the y-intercept is 8.
• The slope \(m\) is crucial in more complex problems where the line has a rise over run, but in this particular case, it's zero since it is a horizontal line.

When graphing, this form not only assists in identifying where the line starts on the y-axis, but it also helps in determining how the line proceeds across the graph. The simplicity of the slope-intercept form ensures efficient graphing of linear equations and inequalities.

Boundary Lines

Boundary lines are essential when graphing linear inequalities. They serve as a visual demarcation between the solutions and non-solutions of an inequality on the graph. In the inequality \(y < 8\), the boundary line is \(y = 8\), and it is dashed, indicating that it is not part of the solution.Here's what to keep in mind about boundary lines:

• An inequality like \(y < 8\) means we draw a dashed line because y can be any value less than 8, but not equal to 8. Hence, the line itself is not included in the solution set.
• Conversely, an inequality with \(\leq\) or \(\geq\) would mean drawing a solid boundary line because points on the line are included in the solution.
• Once the boundary line is drawn, shading below or above the line, depending on the inequality, completes the graph. This shading visually represents the solution set.

Boundary lines help differentiate between different parts of the graph and give clarity to the set of solutions an inequality holds.