Math Problem Statement

Solve the inequalities using the graph of y = f(x) and write the solution sets in interval notation for the following: (a) f(x) < 0 (b) f(x) ≤ 0 (c) f(x) > 0 (d) f(x) ≥ 0

Solution

The image shows a graph y=f(x)y = f(x) with related questions asking to solve inequalities and write solution sets in interval notation. Let me analyze the graph and assist you in solving each part.

Analyzing the Graph:

The graph appears to show intervals where f(x)f(x) is positive, negative, or zero. Key points include where the curve crosses the x-axis, as this helps determine intervals for each inequality.


(a) f(x)<0f(x) < 0:

This represents where the graph is below the x-axis.

  • From the graph, f(x)<0f(x) < 0 on:
    • (,3)(-\infty, -3) and (0,2)(0, 2).

Solution Set: (,3)(0,2)(-\infty, -3) \cup (0, 2).


(b) f(x)0f(x) \leq 0:

This includes where f(x)f(x) is negative or zero (touching the x-axis).

  • From the graph, f(x)0f(x) \leq 0 on:
    • (,3](-\infty, -3] and [0,2][0, 2].

Solution Set: (,3][0,2](-\infty, -3] \cup [0, 2].


(c) f(x)>0f(x) > 0:

This represents where the graph is above the x-axis.

  • From the graph, f(x)>0f(x) > 0 on:
    • (3,0)(-3, 0) and (2,)(2, \infty).

Solution Set: (3,0)(2,)(-3, 0) \cup (2, \infty).


(d) f(x)0f(x) \geq 0:

This includes where f(x)f(x) is positive or zero.

  • From the graph, f(x)0f(x) \geq 0 on:
    • [3,0][-3, 0] and [2,)[2, \infty).

Solution Set: [3,0][2,)[-3, 0] \cup [2, \infty).


Would you like a detailed explanation of how I derived these intervals? Let me know!

Related Questions:

  1. How do you determine the intervals where a function is positive or negative?
  2. What is the significance of using brackets ([][ ]) vs. parentheses (()( )) in interval notation?
  3. How do we check the behavior of a function around critical points like x=3,0,2x = -3, 0, 2?
  4. What changes if the inequality includes or excludes the equals sign (,<\leq, <)?
  5. Can you explain how to solve such inequalities algebraically if given f(x)f(x) explicitly?

Tip: Always check the graph thoroughly for key points where f(x)=0f(x) = 0, as these divide the intervals for inequalities.

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Math Problem Analysis

Mathematical Concepts

Graph Analysis
Inequalities
Interval Notation

Formulas

Understanding interval notation to represent solutions.

Theorems

Basic principles of graph-based inequality solutions.

Suitable Grade Level

Grades 9-12