A graphing calculator is recommended.
Solve the given equation or inequality graphically.
(a) [tex]x^3-4 x-1=0[/tex]
(Enter your answers as a comma-separated list. Round your answers to two decimal places.)
[tex]x =[/tex] $\square$
(b) [tex]x^2-9 \leq|x+3|[/tex] (Enter your answer using interval notation.)
$\square$
Answer (1)
Find the roots of x 3 − 4 x − 1 = 0 graphically: x = − 1.86 , − 0.25 , 2.12 .
Solve x 2 − 9 ≤ ∣ x + 3∣ graphically.
Identify the intersection points of x 2 − 9 and ∣ x + 3∣ : x = − 3 and x = 4 .
Express the solution in interval notation: [ − 3 , 4 ] .
Explanation
Problem Analysis We are given two problems: (a) Solve the equation x 3 − 4 x − 1 = 0 graphically and round the answers to two decimal places. (b) Solve the inequality x 2 − 9 ≤ ∣ x + 3∣ graphically and express the solution in interval notation.
Solving (a) For (a), we need to find the roots of the cubic equation x 3 − 4 x − 1 = 0 . Graphically, these are the x-intercepts of the function f ( x ) = x 3 − 4 x − 1 . Using a graphing calculator or a numerical method, we find the roots to be approximately -1.86, -0.25, and 2.12.
Solving (b) For (b), we need to solve the inequality x 2 − 9 ≤ ∣ x + 3∣ . Graphically, we need to find the intervals where the graph of g ( x ) = x 2 − 9 is below or equal to the graph of h ( x ) = ∣ x + 3∣ . By graphing these two functions, we find that the intersection points are approximately x = -3 and x = 4. The inequality holds for x in the interval [-3, 4].
Final Answer Therefore, the solutions are: (a) x = − 1.86 , − 0.25 , 2.12 (b) x ∈ [ − 3 , 4 ]
Examples
Understanding how to solve equations and inequalities graphically is useful in many real-world situations. For example, engineers might use these techniques to determine the stability of a structure or to optimize the performance of a system. Economists might use them to model supply and demand curves and to find equilibrium points. In general, graphical solutions provide a visual way to understand the behavior of functions and to find approximate solutions to problems that are difficult to solve analytically. For instance, when designing a bridge, engineers use equations to model the forces acting on the structure. Solving these equations graphically helps them visualize the stress distribution and ensure the bridge's stability under various loads.
