Problem 56 - Algebra and Trigonometry

Question

Find the solutions of the inequality by drawing appropriate graphs. State each answer correct to two decimals. $$ (x+1)^{2} \leq x^{3} $$

Step-by-Step Solution

Answer

The solution is \([-1, 1]\).

Step 1: Set Up the Inequality

We start with the inequality \((x + 1)^2 \leq x^3\). This inequality can be rewritten as an equation for finding the boundary values of \(x\): \((x + 1)^2 = x^3\).

Step 2: Solve the Equation

Simplify the equation \((x + 1)^2 = x^3\) by expanding the left side to get \(x^2 + 2x + 1 = x^3\). Rearrange this to form a cubic equation: \(x^3 - x^2 - 2x - 1 = 0\).

Step 3: Factor the Cubic Equation

Using factorization or synthetic division, find that one of the roots of \(x^3 - x^2 - 2x - 1 = 0\) is \(x = 1\). Factor the equation as \((x - 1)(x^2 + 2x + 1) = 0\). Further simplify to \((x - 1)(x + 1)^2 = 0\).

Step 4: Find Roots

The roots of the factorized equation \((x - 1)(x + 1)^2 = 0\) are \(x = 1\) and \(x = -1\). These are the critical points.

Step 5: Analyze the Sign

To solve the inequality \((x + 1)^2 \leq x^3\), test intervals around the roots: \(x < -1\), \(-1 < x < 1\), and \(x > 1\). Evaluating the inequality, we find the solution set where the inequality holds true.

Step 6: Verify the Solution

Determine if the solutions \(x = -1\) and \(x = 1\) satisfy the inequality \((x + 1)^2 \leq x^3\). Substituting, both sides are equal for these values, meaning \(x = -1\) and \(x = 1\) are included.

Step 7: Final Solution

Combining the above results, the solution to the inequality is the interval \([-1, 1]\). Include both ends since the inequality is \(\leq\).

Key Concepts

Inequality Graphing

Graphing inequalities can provide a visual representation that helps us understand where one function is less than or equal to another. In this exercise, you begin by analyzing the inequality \((x + 1)^2 \leq x^3\).

First, identify the functions involved:

• \((x + 1)^2\) — a quadratic function that forms a parabolic curve.
• \(x^3\) — a cubic function that forms an S-shaped curve, typically crossing the x-axis up to three times.

By graphing these, we can observe the intervals on the x-axis where the value of the quadratic function is less than or equal to the cubic function. Notice at points where the two graphs intersect; these are the boundary points of the inequality.

Once graphed, the critical regions will appear between the intersection points, thus confirming the solution intervals.

Cubic Equations

Cubic equations, like the one found in this problem, \[x^3 - x^2 - 2x - 1 = 0, \] have up to three real roots. Solving these requires either factoring, using methods like synthetic division, or applying numerical approximations if factoring is complex.

In this solution, factoring allows us to break down the cubic polynomial. We initially find a single root through trial and error, synthetic division, or the Rational Root Theorem. Once a root like \( x = 1 \) is discovered, it can be factored from the cubic equation, reducing it to simpler terms.

Then, keep factoring or utilize the quadratic formula for any remaining polynomial to find all roots. In this case, the cubic reduced cleanly into multiple linear factors, leading to the roots \( x = 1 \) and \( x = -1 \). Each root represents a point where the inequality changes behavior.

Factoring Equations

Factoring is a crucial tool in simplifying and solving equations. In our exercise, the cubic equation was factored into two parts: \((x - 1)\) and \((x + 1)^2\).

The first step in factoring involves finding one of the roots. Use trial and error to test small numbers or utilize systematic approaches like synthetic division. Once a root is identified, factor it out to simplify the polynomial.

For the remaining portion, consider special patterns or use the quadratic formula if it's a trinomial.

• For example, \((x + 1)^2\) is already in a squared form, recognizable as a perfect square.

By decomposing the equation into these factors, you simplify the complexity of the original cubic expression, making it easier to solve for inequality or other algebraic situations.