Problem 189

Question

$$ x(x-1)^{2} \leq 0 $$

Step-by-Step Solution

Verified

Answer

The inequality $x(x-1)^{2} \leq 0$ is true when $0 \leq x \leq 1$.

1Step 1: Factorize the expression

We are given the inequality: $x(x-1)^{2} \leq 0$ The expression is already factorized.

2Step 2: Find the critical points

To find the critical points, set each factor to zero: $x = 0$ $(x-1) = 0$ or $x=1$ The critical points are $x = 0$ and $x = 1$.

3Step 3: Test the intervals

Now we test each interval between our critical points to determine where the inequality is true: 1. When $x < 0$, we have $x(x-1)^{2} > 0$. (False) 2. When $0 < x < 1$, we have $x(x-1)^{2} < 0$. (True) 3. When $x > 1$, we have $x(x-1)^{2} > 0$. (False) From the above analysis, we can see that the given inequality is true when $0 < x < 1$.

4Step 4: Write the solution

The solution to the inequality is: $0 \leq x \leq 1$ Since the inequality includes the equal sign, both the critical points $0$ and $1$ are included in the solution.

Key Concepts

Critical PointsFactorizationInterval Testing

Critical Points

In mathematics, critical points are values where functions achieve notable events, such as changes from increasing to decreasing. To find the critical points of an inequality like $x(x-1)^2 \leq 0$, we set each factor in the expression to zero. This process helps us identify where the function is zero, providing boundaries between different behaviors of the function.
For this particular inequality, the factors are $x$ and $(x-1)^2$. Solving these gives us the critical points $x = 0$ and $x = 1$. Here is why both values are critical:

For $x = 0$, the expression is zero because any number multiplied by zero is zero.
For $x = 1$, since $(x-1) = 0$, we also have a zero outcome due to a zero factor.

These points help establish the intervals which we will test next to find where the original inequality holds true.

Factorization

Factorization is a process that simplifies expressions by rewriting them as a product of simpler expressions or factors. For the inequality $x(x-1)^2 \leq 0$, it is already in a factorized form. This means we don’t need any more algebraic manipulation to simplify the expression before solving the inequality. Understanding factorization is key because it aids in solving equations and inequalities derived from polynomials.
Here’s why factorization is valuable in this context:

It allows us to easily identify critical points, since setting each factor to zero helps find where the entire expression is zero.
It gives us simpler terms to work with when determining the sign of the expression over different intervals.

Thus, sometimes a given expression is already in a desirable form, sparing us the work of factorizing.

Interval Testing

Interval testing is an essential technique in solving inequalities. Once we identify the critical points, the next step is to test the intervals set between them. These intervals tell us where the inequality is true by checking the sign of the expression within each interval between critical points.
For the inequality $x(x-1)^2 \leq 0$:

We broke the number line into three intervals based on the critical points: $x < 0$, $0 < x < 1$, and $x > 1$.
Testing points from each interval reveals the sign of the expression. For example, choosing a point like $x = 0.5$ in the interval $0 < x < 1$ and substituting back verifies that the inequality holds because $x(x-1)^2$ turns negative.
In the other intervals, test points produce positive expressions, which do not satisfy the inequality.

The result is that the inequality is true only in the interval $0 \leq x \leq 1$. Therefore, interval testing gives us insight into where the solutions of an inequality exist along the number line.

Problem 188

Problem 190

Other exercises in this chapter

Problem 187

$$ x^{2}+2 \leq 3 x \leq 2 x^{2}-5 \text { . } $$

$$ (2-x)(3 x+1)(2 x-3)>0 $$

View solution

Problem 191

$$ (3 x-2)(x-3)^{3}(x+1)^{3}(x+2)^{4}

View solution