Problem 293 - Elementary Algebra

Question

In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} 2 x+2 y>-4 \\ -x+3 y \geq 9 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution is the region where the shaded areas above \( y = -x - 2 \) and \( y \geq \frac{1}{3}x + 3 \) overlap.

Step 1: Rewrite the Inequalities

First, rewrite each inequality in slope-intercept form (y = mx + b). For the first inequality, start with: \[ 2x + 2y > -4 \] Subtract 2x from both sides: \[ 2y > -2x - 4 \] Divide by 2: \[ y > -x - 2 \] For the second inequality, start with: \[ -x + 3y \geq 9 \] Add x to both sides: \[ 3y \geq x + 9 \] Divide by 3: \[ y \geq \frac{1}{3}x + 3 \]

Step 2: Graph the Boundary Lines

Graph the boundary lines for each inequality. The boundary line for \( y > -x - 2 \) is \( y = -x - 2 \), and since it is a 'greater than' inequality, use a dashed line. The boundary line for \( y \geq \frac{1}{3}x + 3 \) is \( y = \frac{1}{3}x + 3 \), and since it is a 'greater than or equal to' inequality, use a solid line.

Step 3: Shade the Solution Regions

Determine where to shade on the graph. For \( y > -x - 2 \), shade above the dashed line. For \( y \geq \frac{1}{3}x + 3 \), shade above the solid line.

Step 4: Find the Intersection Region

The solution to the system of inequalities is the region where the shaded areas overlap. Identify this intersection region on the graph.

Key Concepts

linear inequalities

Linear inequalities are similar to linear equations, but instead of an equal sign, they use inequality symbols such as >, <, ≥, or ≤. They form a region on a graph rather than a single line. Solving systems of linear inequalities involves finding the areas that satisfy all given inequalities.
Solving these inequalities helps to understand how different constraints affect possible solutions.

slope-intercept form

The slope-intercept form is a useful way to write the equation of a line, given by\:\( y = mx + b \). Here, 'm' represents the slope, and 'b' represents the y-intercept.
To graph inequalities, first convert them into slope-intercept form. For example, for\: \( 2x + 2y > -4 \):

• Subtract \: 2x \: from both sides to get \: \( 2y > -2x - 4 \),
• Then divide by \: 2 \: to get \: \( y > -x - 2 \).

This transformation helps in graphing the boundary lines.

graphing boundary lines

Boundary lines are crucial when graphing inequalities. They represent the equality part of the inequality. For instance, in \: \( y > -x - 2 \), the boundary line is \: \( y = -x - 2 \). Determine the type of line:

• Dashed line: Use it for inequalities like \: \( > \) \: or \: \( < \), indicating the points on the line are not included.
• Solid line: Use it for \: \( \geq \) \: or \: \( \leq \) \: to show that the points on the line are included.

Accurately drawing these lines is key to correctly interpreting the solution.

shading solution regions

Shading is the final step in graphing inequalities. It helps to visualize the solution set on the graph. Once boundary lines are plotted:
* For \: \( y > -x - 2 \): Shade the area above the dashed line, showing all the y-values greater than \: \( -x - 2 \).
* For \: \( y \geq \frac{1}{3}x + 3 \): Shade the area above the solid line, indicating the y-values that meet or exceed \: \( \frac{1}{3}x + 3 \).
The solution region for the system of inequalities is the overlapping area of the shaded regions. This intersection region represents all the points that satisfy both inequalities simultaneously.