Question

A solid line, a dashed line, and three points are graphed on a coordinate plane. The x-axis ranges from negative 7 to 10 in increments of 1. The y-axis ranges from negative 5 to 7 in increments of 1.

Which of the points shown on the graph is a solution to the simultaneous inequalities y>3x+1
and y≤2x+5
? Write the x-value first and the y-value second.
the coordinates:
-5,5
-1,2
9,5

Answer 1

To determine which of the given points is a solution to the simultaneous inequalities \( y > 3x + 1 \) and \( y \leq 2x + 5 \), we will evaluate each point against both inequalities.

1. **Point (-5, 5)**:
- For \( y > 3x + 1 \):
\[
5 > 3(-5) + 1 \implies 5 > -15 + 1 \implies 5 > -14 \quad \text{(True)}
\]
- For \( y \leq 2x + 5 \):
\[
5 \leq 2(-5) + 5 \implies 5 \leq -10 + 5 \implies 5 \leq -5 \quad \text{(False)}
\]

2. **Point (-1, 2)**:
- For \( y > 3x + 1 \):
\[
2 > 3(-1) + 1 \implies 2 > -3 + 1 \implies 2 > -2 \quad \text{(True)}
\]
- For \( y \leq 2x + 5 \):
\[
2 \leq 2(-1) + 5 \implies 2 \leq -2 + 5 \implies 2 \leq 3 \quad \text{(True)}
\]

3. **Point (9, 5)**:
- For \( y > 3x + 1 \):
\[
5 > 3(9) + 1 \implies 5 > 27 + 1 \implies 5 > 28 \quad \text{(False)}
\]
- For \( y \leq 2x + 5 \):
\[
5 \leq 2(9) + 5 \implies 5 \leq 18 + 5 \implies 5 \leq 23 \quad \text{(True)}
\]

After evaluating all the points:

- Point (-5, 5) does not satisfy the second inequality.
- Point (-1, 2) satisfies both inequalities.
- Point (9, 5) does not satisfy the first inequality.

Therefore, the point that is a solution to the simultaneous inequalities is **(-1, 2)**.