The system of linear inequalities is as follows:

1. $y \leq 2x + 5$
2. $y < -\frac{3}{2}x + 5$

To find a point that satisfies this system, we need to identify a region where both inequalities overlap. We can do this by looking for a point that satisfies both conditions.

Step-by-step approach:

1. The first inequality, $y \leq 2x + 5$, represents a region below or on the line $y = 2x + 5$.
2. The second inequality, $y < -\frac{3}{2}x + 5$, represents a region below the line $y = -\frac{3}{2}x + 5$.

These lines will intersect at some point, and the solution must lie in the area where both inequalities overlap. You can solve for the intersection point of the two lines by setting the right-hand sides of the equations equal to each other:

$2x + 5 = -\frac{3}{2}x + 5$

Solving for $x$:

1. Subtract 5 from both sides: $2x = -\frac{3}{2}x$

2. Add $\frac{3}{2}x$ to both sides: $2x + \frac{3}{2}x = 0$

3. Convert 2x into a fraction: $\frac{4}{2}x + \frac{3}{2}x = 0$

4. Combine like terms: $\frac{7}{2}x = 0$

5. Solve for $x$: $x = 0$

Substituting $x = 0$ back into either equation to find $y$:

Using $y = 2x + 5$: $y = 2(0) + 5 = 5$

Thus, the two lines intersect at the point $(0, 5)$.

Check the inequalities:

• For $y \leq 2x + 5$: At $(0, 5)$, we have $5 \leq 2(0) + 5$, which is true.
• For $y < -\frac{3}{2}x + 5$: At $(0, 5)$, we have $5 < -\frac{3}{2}(0) + 5$, which is false, because the inequality is strict.

Since the second inequality does not include the point $(0, 5)$ due to the strict inequality, we need to pick a point strictly below the line $y = -\frac{3}{2}x + 5$ but also satisfy $y \leq 2x + 5$.

Trying another point:

Let’s test $x = 1$ and substitute into both inequalities.

• For the first inequality: $y \leq 2(1) + 5 = 7$.
• For the second inequality: $y < -\frac{3}{2}(1) + 5 = \frac{7}{2} = 3.5$.

If we pick $y = 3$ (since it must be less than 3.5), the point $(1, 3)$ satisfies both inequalities.

Conclusion:

A point that satisfies both inequalities is $(1, 3)$.

Would you like me to explain this further, or do you have any additional questions?

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Here are 5 follow-up questions:

1. How do we graph these inequalities to visually check the solution?
2. How can we solve a system of three linear inequalities?
3. What does the inequality symbol $\leq$ represent geometrically?
4. How do you solve for the intersection of two linear inequalities algebraically?
5. How would the solution change if the second inequality was $y \leq -\frac{3}{2}x + 5$?

Tip: When dealing with strict inequalities like $<$, make sure the solution does not include the boundary line, while for non-strict inequalities like $\leq$, the boundary line is included.