Problem 46 - Algebra 2

Question

Solve each system of inequalities by graphing. $$ \begin{array}{l}{2 y+x \geq 4} \\ {y \geq x-4}\end{array} $$

Step-by-Step Solution

Answer

The solution is the region above both lines, including both solid lines.

Step 1: Graph the inequality \(2y+x \geq 4\)

Firstly, rearrange the inequality in slope-intercept form (\(y = mx + b\)). To do so, isolate \(y\): \[2y \geq -x + 4\] Divide the entire inequality by 2: \[y \geq -\frac{1}{2}x + 2\] Now plot the line \(y = -\frac{1}{2}x + 2\) using its slope (-1/2) and y-intercept (2). Since the inequality sign is \(\geq\), the area above the line is shaded including the line itself (drawn as a solid line).

Step 2: Graph the inequality \(y \geq x-4\)

For the second inequality, it is already in the slope-intercept form \(y = mx + b\). The equation is \(y = x - 4\) with a slope of 1 and a y-intercept of -4. Plot this line on the graph. Since \(y\) is greater than or equal to \(x - 4\), shade the region above the line, including the line itself (represented as a solid line).

Step 3: Find the solution region

The solution to the system of inequalities is the region where the shaded areas from both inequalities overlap. Region above the first line (\(y \geq -\frac{1}{2}x + 2\)) and above the second line (\(y \geq x - 4\)) is shaded. The overlapped shaded area is the solution region, which is the portion of the coordinate plane that satisfies both inequalities.

Key Concepts

Graphing Inequalities

When graphing inequalities, the first step is to transform each inequality into an equation that represents a line. You plot this line on a coordinate plane to visualize the boundary of the inequality. If the inequality includes equality, like ">=" or "<=", you draw a solid line to indicate that points on the line are part of the solution. If the inequality is strict, like ">" or "<", a dashed line is used.
Once the line is graphed, you shade the region of the plane that expresses the solution to the inequality. This is done by testing a point not on the line. For example, checking (0,0) often helps to determine which side of the line to shade. If (0,0) does not satisfy the inequality, the opposite side from the origin is shaded.

Slope-Intercept Form

The slope-intercept form of a linear equation is wonderfully straightforward: it's expressed as \(y = mx + b\). Here, \(m\) represents the slope, and \(b\) is the y-intercept.

• **Slope (m):** This is the rate of change of the line. It tells you how steep the line is. A positive slope means the line rises as it moves from left to right, while a negative slope means the line falls.
• **Y-intercept (b):** This is where the line crosses the y-axis. It's the point on the line where x is zero.

To graph an inequality, you first rearrange it into the slope-intercept form so you can easily identify the slope and y-intercept. This helps in plotting the line correctly on a graph before determining which side to shade for the inequality part.

Solution Region

The solution region of a system of inequalities is where the conditions of all inequalities are satisfied simultaneously. This region appears as the overlapped area when graphing more than one inequality. To find it, you first graph each inequality on the same coordinate plane, following the shading rules for each. The solution area is the intersection of these individual shaded regions.
Once you have graphed all the equations, the solution region will be evident as the area where all shaded sections overlap. Points within this region are solutions to the inequality system, meaning they satisfy all equations in the system when substituted into them.

Coordinate Plane

Understanding the coordinate plane is essential for graphing inequalities effectively. A coordinate plane is a two-dimensional space formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical).
Each point on this plane has two coordinates: an x-coordinate and a y-coordinate, generally written as (x, y).

• The **x-axis** is the horizontal line that divides the plane into upper and lower halves.
• The **y-axis** is the vertical line that divides the plane into left and right halves.
• The **origin (0,0)** is where the x-axis and y-axis meet. It serves as a reference point for finding all other points on the plane.

When graphing, you locate points based on these coordinates, and understanding the layout of the coordinate plane helps in accurately plotting lines and solution regions.