https://file.aiquickdraw.com/mathbot/file/1726210551127xlocv1ik.jpg

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 \( -6 \leq x < 5 \).

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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.

Solve the equation f(x) = g(x) and the inequality g(x) ≤ f(x) < h(x) using the graph.

This problem involves solving the equation f(x) = g(x) using a graph and determining where the inequality g(x) ≤ f(x) < h(x) holds. The solution covers graphing concepts and is suitable for students in Grades 9-11.

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