Sure! Here's an example of a system of two linear inequalities:
- \(2x + 3y < 6\)
- \(x - y \geq 1\)
Graphing the System
-
To graph these inequalities, you would first graph the corresponding equations:
-
For \(2x + 3y = 6\), you can find the intercepts. The x-intercept occurs when \(y = 0\): \[ 2x + 3(0) = 6 \implies x = 3 \quad \text{(point (3,0))} \] The y-intercept occurs when \(x = 0\): \[ 2(0) + 3y = 6 \implies y = 2 \quad \text{(point (0,2))} \] This line is dashed because the inequality is strict (\(<\)).
-
For \(x - y = 1\), the x-intercept occurs when \(y = 0\): \[ x - 0 = 1 \implies x = 1 \quad \text{(point (1,0))} \] The y-intercept occurs when \(x = 0\): \[ 0 - y = 1 \implies y = -1 \quad \text{(point (0,-1))} \] This line is solid because the inequality is non-strict (\( \geq \)).
-
Solution Region
The solution to the system of inequalities consists of the points that satisfy both inequalities simultaneously. You would shade the appropriate regions for each inequality on the graph.
- For \(2x + 3y < 6\), shade below the dashed line.
- For \(x - y \geq 1\), shade above or on the solid line.
The overlapping shaded region represents the solution set for the system of linear inequalities.
If you'd like further details or explanations, feel free to ask!
1 answer