Which is the graph of the system $x+3y > -3$ and $y < \frac{1}{2}x + 1$?

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06-11-2026
Mathematics

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Which is the graph of the system $x+3y > -3$ and $y < \frac{1}{2}x + 1$?

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-06T17:38:09-05:00

Convert inequalities to slope-intercept form: $y > -\frac{1}{3}x - 1$ and $y < \frac{1}{2}x + 1$.
Recognize that both lines are dashed because the inequalities are strict.
Locate the intersection point at $(-2.4, -0.2)$ by solving the system of linear equations.
The final graph consists of the region above the line $y = -\frac{1}{3}x - 1$ and below the line $y = \frac{1}{2}x + 1$, with boundary lines omitted.
The final answer describes the region $\boxed{y > -\frac{1}{3}x - 1 \text{ and } y < \frac{1}{2}x + 1}$.

Explanation

Analyzing the Inequalities

We are given a system of two linear inequalities:

$x + 3y > -3$
$y < \frac{1}{2}x + 1$

To analyze these, we first convert them into slope-intercept form ($y = mx + b$).
For the first inequality:
\[3y > -x - 3\]
\[y > -\frac{1}{3}x - 1\]
This represents the region above the line $y = -\frac{1}{3}x - 1$. Since the inequality is strict (greater than), the boundary line is drawn as a dashed line. This line has a y-intercept of $-1$ and an x-intercept of $-3$.

For the second inequality:
\[y < \frac{1}{2}x + 1\]
This represents the region below the line $y = \frac{1}{2}x + 1$. Because the inequality is strict (less than), the boundary line is also dashed. This line has a y-intercept of $1$ and an x-intercept of $-2$.

Finding the Intersection of Boundary Lines

To find the intersection of these two regions, we first identify the point where the two boundary lines meet. We set the equations equal to each other:
\[-\frac{1}{3}x - 1 = \frac{1}{2}x + 1\]
To solve for $x$, we can multiply the entire equation by $6$ to eliminate fractions:
\[-2x - 6 = 3x + 6\]
Adding $2x$ to both sides:
\[-6 = 5x + 6\]
Subtracting $6$ from both sides:
\[-12 = 5x \Rightarrow x = -2.4\]
Now, substitute $x = -2.4$ back into one of the line equations to find $y$:
\[y = \frac{1}{2}(-2.4) + 1 = -1.2 + 1 = -0.2\]
The lines intersect at the point $(-2.4, -0.2)$.

Describing the Solution Region

The solution to the system of inequalities is the overlapping region where both conditions are satisfied: $y > -\frac{1}{3}x - 1$ and $y < \frac{1}{2}x + 1$.

The line $y = -\frac{1}{3}x - 1$ (dashed) divides the plane, and we shade the region above it.
The line $y = \frac{1}{2}x + 1$ (dashed) divides the plane, and we shade the region below it.

The resulting graphical representation is a region (a sector) formed by the intersection of these two half-planes, with the vertex at $(-2.4, -0.2)$, where the boundary lines are dashed to indicate that points on the lines themselves are not included in the solution set.

Examples

In urban planning, architects might use systems of linear inequalities to define restricted zones for construction. For example, if a park area is restricted by certain environmental buffer lines (represented as linear boundaries), the allowable building zone is the intersection of these inequalities. Visualizing the valid area on a map helps ensure that development remains within safe and legal limits, just as graphing inequalities helps identify the valid region for a system of mathematical equations.