Problem 24 Graph the solution set. $$x-3 ... [FREE SOLUTION]

Chapter 2: Problem 24

Graph the solution set. $$x-3 y>0$$

Short Answer

Expert verified

Graph the line $y = \frac{x}{3}$ as dashed and shade below it.

Step by step solution

Rearrange the Inequality

To graph the inequality, first rearrange it to solve for $y$. Begin with the original inequality: $x - 3y > 0$. To isolate $y$, subtract $x$ from both sides to get: $-3y > -x$. Next, divide through by $-3$, remembering to reverse the inequality sign when dividing by a negative number: $y < \frac{x}{3}$.

Graph the Boundary Line

Graph the boundary line for the equation $y = \frac{x}{3}$. This line has a y-intercept of 0 and a slope of $\frac{1}{3}$. Draw this line as a dashed line because the inequality is strictly less than (<), not less than or equal to ($\leq$).

Determine the Shading Region

Since the inequality is $y < \frac{x}{3}$, the solution set is below the line. To determine this region, pick a test point not on the line, such as (0,0). Substitute this point into the inequality: $0 < \frac{0}{3}$, which is true. Therefore, shade the area below the dashed line as this represents the solution set.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Boundary Line

When graphing an inequality like $x - 3y > 0$, it's important to first identify the boundary line. The boundary line represents the equation when the inequality becomes an equality, $y = \frac{x}{3}$ in this case. This line serves as a clear marker dividing the coordinate plane into regions where the inequality holds true and where it doesn't. Here's how to handle the boundary line:

**Identify the Type of Line**: For an inequality that uses 'greater than' ($>$) or 'less than' ($<$), the boundary line is dashed. This indicates that points on the line itself are not part of the solution set.

**Graphing the Line**: Analyze its slope and intercept. In the line $y = \frac{x}{3}$, the slope is $\frac{1}{3}$, and it passes through the origin, $(0,0)$.

The boundary line visually divides the plane and guides us to determine the region that satisfies the inequality.

Shading Region

Once the boundary line for the inequality $y < \frac{x}{3}$ is drawn, the next step is to determine which side of the line to shade. The shaded region on a graph visually represents all the solutions to the inequality.Here's a helpful approach:

**Pick a Test Point**: Choose a point not on the boundary line, like the origin $(0, 0)$, unless the line passes through it鈥攁ny other point can work in that case.

**Substitute and Test**: Plug this point into the inequality to see if the inequality holds true. For $(0,0)$, substitute into $0 < \frac{0}{3}$. The statement holds true, confirming that this side of the line is the solution area.

**Shade the Region**: Since the test point satisfies the inequality, shade the entire region including it. For $y < \frac{x}{3}$, shade below the line.

The shaded region indicates where all solution points lie, showing a clear visual answer to the inequality problem.

Inequality Solution Set

The inequality solution set consists of all the points on a coordinate plane that satisfy a given inequality. In the case of $x - 3y > 0$, or equivalently $y < \frac{x}{3}$, every point in the shaded region adheres to the inequality's condition.Key characteristics of an inequality solution set include:

**Exclusivity of the Boundary**: Because the original inequality is strict ($>$ or $<$), points on the dashed boundary line do not belong to the solution set. This distinction is crucial for determining the accuracy of the set.

**Continuous Nature**: The solution set is often a continuous region. This means there are infinitely many solutions forming a continuous area rather than discrete points.

**Visual Representation**: The solution set is best understood through its visual representation on a graph. Seeing the shaded region in relation to the boundary line helps one comprehend the extent and range of solutions.

Identifying the inequality solution set via graphing helps visualize complex problems, providing clarity on where solutions reside.

91影视

Graph the solution set. $$x-3 y>0$$

Short Answer

Step by step solution

Rearrange the Inequality

Graph the Boundary Line

Determine the Shading Region

Key Concepts

Boundary Line

Shading Region

Inequality Solution Set

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