Problem 57 - Algebra 2

Question

Graph each inequality. \(3 x+9 y \geq-15\)

Step-by-Step Solution

Answer

Plot the line \(3x + 9y = -15\) with intercepts \((-5, 0)\) and \((0, -\frac{5}{3})\), and shade above it.

Step 1: Rewrite the Inequality as an Equation

First, rewrite the inequality as an equation to identify the boundary line. The inequality is given as \(3x + 9y \geq -15\). So, we convert it to the equation \(3x + 9y = -15\).

Step 2: Find the Intercepts

To graph the boundary line, we need to find the x-intercept and y-intercept. - For the x-intercept, set \(y = 0\) in the equation: \(3x + 9(0) = -15\) which simplifies to \(3x = -15\), so \(x = -5\).- For the y-intercept, set \(x = 0\) in the equation: \(3(0) + 9y = -15\) which simplifies to \(9y = -15\), so \(y = -\frac{5}{3}\).

Step 3: Plot the Intercepts and Draw the Boundary Line

Use the intercepts \((-5, 0)\) and \((0, -\frac{5}{3})\) to draw the boundary line on a graph. This line will either be solid or dashed. Since our inequality is \(\geq\) (greater than or equal to), the line is solid.

Step 4: Determine the Shading Area

Identify which side of the boundary line to shade. Use a test point, like \((0, 0)\), which is not on the line.- Substitute \((0, 0)\) into the inequality: \(3(0) + 9(0) \geq -15\) simplifying to \(0 \geq -15\), which is true.- This true result means the region containing the point \((0, 0)\) should be shaded, indicating the solution set.

Key Concepts

Linear Inequality

Graphing a linear inequality involves converting an inequality into a visual representation on a coordinate plane. Unlike linear equations, which use an equals sign, linear inequalities use inequality signs like ">", "<", "≥", or "≤". This change reflects that the inequality encompasses not just a line, but a region of the plane. Linear inequalities are fundamental in understanding regions of possible solutions.

To graph a linear inequality, follow these steps:

• Rewrite the inequality as an equation to easily find the boundary line.
• Determine intercepts, which are key points that define the boundary line.
• Decide whether this line is solid or dashed, depending on whether the inequality includes (≥, ≤) or excludes (>, <) the line itself.
• Finally, shade the area where the inequality holds true, indicating all solutions are in that region.

Boundary Line

A boundary line is a critical component of graphing linear inequalities, essentially acting as a border between solution and non-solution regions. To determine it, one must first rewrite the inequality as an equation. For example, changing the inequality \(3x + 9y \geq -15\) to the equation \(3x + 9y = -15\) reveals the boundary.

This boundary line can either be solid or dashed:

• Solid Line: Used when the inequality includes the boundary itself (≥, ≤). It shows that points on the line are part of the solution set.
• Dashed Line: Used when the inequality does not include the boundary (>, <). Points on the line are not part of the solution set.

The boundary line aids in visual judgment by clarifying which region satisfies the inequality based on test points.

X-Intercept

The x-intercept is where your graph crosses the x-axis. This is a vital point to find when plotting the boundary line as it helps to establish one of the two points needed for drawing the line correctly. In simple terms, the x-intercept occurs when the value of y is zero.

To find the x-intercept of the boundary line obtained from the equation \(3x + 9y = -15\), plug in 0 for \(y\):

• Set the equation as: \(3x + 9(0) = -15\).
• Simplify it to \(3x = -15\).
• Solve for \(x\) by dividing both sides by 3, resulting in \(x = -5\).

Thus, the x-intercept is \((-5, 0)\), providing a starting point for constructing the boundary line.

Y-Intercept

The y-intercept is the point where the graph intersects the y-axis, vital for locating part of the boundary line. Along with the x-intercept, it enables one to draw the boundary line accurately on the graph. The value of \(x\) at the y-intercept is zero.

For our equation \(3x + 9y = -15\), to find the y-intercept, insert 0 for \(x\):

• Rewrite as: \(3(0) + 9y = -15\).
• This simplifies to \(9y = -15\).
• Divide both sides by 9 to solve for \(y\): \(y = -\frac{5}{3}\).

Hence, the y-intercept is \((0, -\frac{5}{3})\), completing the necessary points to plot the line accurately.