Which is the graph of $4 x+2 y<3$?

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06-11-2026
Mathematics

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Which is the graph of $4 x+2 y<3$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T09:40:51-05:00

Find the boundary line by solving $4x + 2y = 3$, which has a y-intercept of $1.5$, an x-intercept of $0.75$, and a slope of $-2$.
Recognize that the strict inequality $<$ indicates a dashed boundary line.
Test the origin $(0,0)$ to find that the region including the origin is the correct area to shade.
Conclude that the graph should feature a dashed line passing through $(0, 1.5)$ and $(0.75, 0)$, with the side containing the origin shaded.

Explanation

Analyzing the Inequality and its Boundary Line

The problem asks to determine the graph of the inequality $4x + 2y < 3$. To identify the correct graphical representation, we first examine the boundary line defined by the corresponding linear equation: $4x + 2y = 3$. This line serves as the edge of the shaded region in the Cartesian coordinate system.

Determining Intercepts and Slope

To determine the behavior of the line, we calculate its intercepts. Setting $x = 0$ allows us to find the y-intercept: $4(0) + 2y = 3 \implies 2y = 3 \implies y = 1.5$. Thus, the line passes through $(0, 1.5)$. Setting $y = 0$ allows us to find the x-intercept: $4x + 2(0) = 3 \implies 4x = 3 \implies x = 0.75$. Thus, the line passes through $(0.75, 0)$. Rewriting the equation in slope-intercept form $y = mx + b$, we have $2y = -4x + 3 \implies y = -2x + 1.5$. The slope is $-2$, and the y-intercept is $1.5$.

Identifying Graph Properties and Shaded Region

Since the given inequality $4x + 2y < 3$ uses the strict inequality symbol ('<'), the boundary line is not part of the solution set. Therefore, it must be represented as a dashed line. Furthermore, we must determine which side of the line to shade. Testing the origin $(0, 0)$, we substitute it into the inequality: $4(0) + 2(0) < 3 \implies 0 < 3$. Since this statement is true, the region containing the origin should be shaded.

Conclusion

The graph of the inequality $4x + 2y < 3$ is defined by a dashed line passing through $(0, 1.5)$ and $(0.75, 0)$ with a slope of $-2$. The shaded region lies below and to the left of this line, including the origin. Any graph satisfying these conditions is the correct representation.

Examples

Understanding how to graph inequalities is essential in fields like linear programming, which is used to optimize resource allocation. For example, if a factory has a limited amount of two raw materials (represented by $x$ and $y$), the constraints on production can be modeled as inequalities. Graphing these inequalities helps identify the "feasible region" of production levels that satisfy all constraints, ensuring efficient operation without exceeding available resources.