Let's break down the information provided in the image to solve the questions.

Graph analysis:

1. $y = f(x)$ is the slanted line that passes through points $(-6, -5)$ and $(5, 12)$.
2. $y = g(x)$ is the horizontal line that passes through $(0, -5)$ and extends horizontally.
3. $y = h(x)$ is the horizontal line that passes through $(0, 12)$ and extends horizontally.

(a) Solve the equation $f(x) = g(x)$:

This means we are looking for the $x$-coordinate where the lines $y = f(x)$ and $y = g(x)$ intersect.

• From the graph, we observe that the lines intersect at point $(0, -5)$.
• Thus, the solution to $f(x) = g(x)$ is $x = 0$.

(b) Solve the inequality $g(x) \leq f(x) < h(x)$:

This inequality involves finding the values of $x$ where the following conditions are true:

1. $g(x) \leq f(x)$ — The line $g(x)$ is below or equal to the line $f(x)$.
2. $f(x) < h(x)$ — The line $f(x)$ is below the line $h(x)$.

From the graph:

• $g(x)$ is constant at $-5$, and $f(x)$ is the slanted line, so we need to check where $f(x) \geq -5$ and $f(x) < 12$.
• From the graph, we observe that the inequality holds for $-6 \leq x \leq 5$, since:
  • $f(x) = g(x) = -5$ at $x = 0$, and for values from $x = -6$ to $x = 5$, $f(x)$ rises from $-5$ to $12$, but is strictly less than $12$ until $x = 5$.

Thus, the solution for part (b) is $-6 \leq x < 5$.

Would you like more details on any of the solutions or clarification on a specific point?

Related Questions:

1. How do you derive the equation of a line passing through two points?
2. Can you explain the meaning of graphing an inequality on a number line?
3. How would the solution change if the line for $h(x)$ was not horizontal?
4. What would happen if the lines $f(x)$ and $g(x)$ were parallel?
5. How do you solve inequalities graphically for more complex curves?

Tip:

When solving equations graphically, intersections represent the points where the functions are equal. Inequalities can be solved by identifying the regions where one graph lies above or below another.