Question

which is true about the solution to the system of inequalities shown?
y ≥ 3x + 1
y ≤ 3x − 3
options:

• only values that satisfy y ≥ 3x + 1 are solutions.
• only values that satisfy y ≤ 3x − 3 are solutions.
• values that satisfy either y ≥ 3x + 1 or y ≤ 3x − 3 are solutions.
• there are no solutions.

Explanation:

Step1: Analyze line slopes

Both inequalities have the form $y = mx + b$, where $m=3$ for both. This means the lines are parallel (same slope, never intersect).

Step2: Compare y-intercepts

The first inequality has a y-intercept $b_1=1$, the second has $b_2=-3$. The line $y=3x+1$ lies entirely above $y=3x-3$.

Step3: Evaluate inequality overlap

We need $y \geq 3x+1$ (region above the upper line) and $y \leq 3x-3$ (region below the lower line). There is no overlap between these two regions.

Answer:

There are no solutions.