Let's solve the inequalities one by one using the provided graphs:

a) $x^2 - 10x + 25 > 0$

This corresponds to $f(x) = x^2 - 10x + 25$.

• The equation factors to $(x - 5)^2 > 0$.
• Observing the graph, $f(x)$ touches the x-axis at $x = 5$, indicating a double root.
• The inequality is strict ($> 0$), so $x \neq 5$. The solution is:

$x \in (-\infty, 5) \cup (5, \infty)$

b) $-x^2 + \frac{5}{2}x + 6 \geq 0$

This corresponds to $h(x) = -x^2 + \frac{5}{2}x + 6$.

• Observing the graph of $h(x)$, it crosses the x-axis at approximately $x = -1$ and $x = 6$.
• $h(x) \geq 0$ where the graph is on or above the x-axis, i.e., $x \in [-1, 6]$.

c) $2x^2 + 2x - 4 < 0$

This corresponds to $g(x) = 2x^2 + 2x - 4$.

• Observing the graph of $g(x)$, it crosses the x-axis at approximately $x = -2.5$ and $x = 0.5$.
• $g(x) < 0$ where the graph is below the x-axis, i.e., $x \in (-2.5, 0.5)$.

d) $2(x+2)(x-1) \geq 0$

The roots of $2(x+2)(x-1)$ are $x = -2$ and $x = 1$.

• Observing the graph (factored form), the parabola changes signs at these roots.
• The inequality $\geq 0$ includes the points where the graph touches the x-axis and is above it. Thus:

$x \in [-2, 1] \cup (1, \infty)$

e) $-(x-3)^2 - 3 < 0$

Rewriting: $-(x-3)^2 - 3 < 0$ becomes $(x-3)^2 > -3$.

• Since $(x-3)^2 \geq 0$ always, this inequality is always true.
• Solution: All real numbers, $x \in (-\infty, \infty)$.

f) $(x-5)^2 \leq 0$

This corresponds to $f(x) = (x-5)^2$.

• $(x-5)^2 \leq 0$ only holds true when $(x-5)^2 = 0$, i.e., $x = 5$.
• Solution: $x = 5$.

Would you like detailed explanations or visualizations for any of these? Let me know!

Related Questions:

1. How do we determine the roots of a quadratic function graphically?
2. Why is the solution to $x^2 > 0$ different from $x^2 \geq 0$?
3. What is the role of the vertex in quadratic inequalities?
4. How can we analyze inequalities without solving them algebraically?
5. What is the impact of negative coefficients on the shape of a parabola?

Tip:

When solving inequalities using graphs, always pay attention to whether the inequality is strict ($>$ or $<$) or non-strict ($\geq$ or $\leq$). This affects the inclusion or exclusion of roots.