Sketch the solution to the system of inequalities: y > 4x - 3, y ≥ -2x + 3
Solution
The image contains a problem from a worksheet titled "Systems of Inequalities." The specific problem provided is:
y &> 4x - 3 \\
y &\geq -2x + 3
\end{aligned}$$
To solve this system of inequalities, we need to graph both inequalities on the same coordinate plane and find the region where both conditions are true.
### Steps to Solve:
1. **Graph the first inequality $$y > 4x - 3$$:**
- First, graph the line $$y = 4x - 3$$. This is a straight line with a slope of 4 and a y-intercept at -3.
- Because the inequality is strict ($$>$$), the line will be dashed, indicating that points on the line are not included in the solution.
- The region above the line represents the solution to this inequality.
2. **Graph the second inequality $$y \geq -2x + 3$$:**
- Graph the line $$y = -2x + 3$$. This is a straight line with a slope of -2 and a y-intercept at 3.
- Since the inequality is non-strict ($$\geq$$), the line will be solid, meaning that points on the line are included in the solution.
- The region above the line represents the solution to this inequality.
3. **Determine the solution region:**
- The solution to the system of inequalities is the region where both shaded areas overlap.
- Since $$y > 4x - 3$$ is above the first line and $$y \geq -2x + 3$$ is above the second line, the overlapping region above both lines is the solution to the system.
Would you like a detailed graph of this solution, or do you have any other questions? Here are five related questions to consider:
1. How do you determine if a line should be dashed or solid in an inequality graph?
2. What does the slope of a line represent in a linear equation?
3. How do you find the point of intersection between two lines?
4. What is the significance of the y-intercept in graphing linear inequalities?
5. How can you test whether a specific point satisfies a system of inequalities?
**Tip:** Always shade the region above or below the line based on the inequality sign: above for $$y >$$ or $$y \geq$$, and below for $$y <$$ or $$y \leq$$.
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Math Problem Analysis
Mathematical Concepts
Algebra Linear Inequalities Graphing Systems of Inequalities
Formulas
Slope-intercept form y = mx + b
Theorems
Properties of inequalities Graphical representation of linear inequalities