Problem 24 - Calculus with Concepts in Calculus

Question

Solve the inequality. $$ (x-1)(x-2)(x-3) \leq 0 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, 1] \cup [2, 3]\).

Step 1: Identify Roots

The roots of the inequality \((x-1)(x-2)(x-3) = 0\) are found by setting each factor equal to zero. This gives us the roots: \(x = 1\), \(x = 2\), \(x = 3\).

Step 2: Determine Intervals

The roots split the number line into intervals. Here, the intervals are \((-\infty, 1)\), \((1, 2)\), \((2, 3)\), and \((3, \infty)\).

Step 3: Test Each Interval

Select a test point from each interval to determine if the product \((x-1)(x-2)(x-3)\) is negative or zero:- Choose \(x = 0\) for \((-\infty, 1)\): \((0-1)(0-2)(0-3) = (-1)(-2)(-3) = -6\), negative.- Choose \(x = 1.5\) for \((1, 2)\): \((1.5-1)(1.5-2)(1.5-3) = (0.5)(-0.5)(-1.5) = 0.375\), positive.- Choose \(x = 2.5\) for \((2, 3)\): \((2.5-1)(2.5-2)(2.5-3) = (1.5)(0.5)(-0.5) = -0.375\), negative.- Choose \(x = 4\)for \((3, \infty)\): \((4-1)(4-2)(4-3) = (3)(2)(1) = 6\), positive.

Step 4: Include Zero Points

The inequality includes the \(\leq\) symbol, so we include the points where \((x-1)(x-2)(x-3) = 0\). Thus, points \(x = 1\), \(x = 2\), and \(x = 3\) are also part of the solution.

Step 5: Write the Solution

Based on steps 3 and 4, the solution to the inequality is the union of the intervals where the product is either zero or negative: \((-\infty, 1] \cup [2, 3]\).

Key Concepts

Roots of Equations

Finding the roots of an equation is like looking for the spots where a road meets the ground. In math terms, roots are the values of \(x\) that make the equation, in this case \((x-1)(x-2)(x-3) = 0\), true. To find these special numbers, we set each part of the product to zero and solve for \(x\).

• For \(x - 1 = 0\), solve and get \(x = 1\).
• For \(x - 2 = 0\), solve and get \(x = 2\).
• For \(x - 3 = 0\), solve and get \(x = 3\).

These numbers, 1, 2, and 3, are the roots of our inequality. They act like turning points that divide a number line into separate intervals. These roots are crucial because they help us understand the overall behavior of the inequality.

Interval Notation

Interval notation is like sending a postcard with destinations marked on it, showing where you begin, end, and any stops along the way. When dealing with inequalities, interval notation helps express the solutions as ranges on a number line. Let’s break it down further. The roots we found earlier — 1, 2, and 3 — divide the number line into several sections, known as intervals. These intervals are:

• \((-\infty, 1)\) for all numbers less than 1.
• \((1, 2)\) for numbers between 1 and 2.
• \((2, 3)\) for numbers between 2 and 3.
• \((3, \infty)\) for numbers greater than 3.

We use '(' and ')' to denote that the endpoints are not included, and '[', ']' to show that they are included in the solution. For our original inequality's solution, we include points where the expression equals zero (\([\) or \(])\), confirming that points 1, 2, and 3 are also included where appropriate.

Polynomial Inequalities

Polynomial inequalities, like the one in our exercise, are like telling a story about what whole sections of \(x\)-values do together. To solve these inequalities, we need to check whether sections of values are positive or negative by using test points. Here's how it goes:After dividing the number line into intervals using the roots, we pick test points from each interval to determine the sign of the expression in that section:

• For \((-\infty, 1)\), pick \(x = 0\) which makes the expression negative.
• For \((1, 2)\), pick \(x = 1.5\) which makes the expression positive.
• For \((2, 3)\), pick \(x = 2.5\) which makes the expression negative.
• For \((3, \infty)\), pick \(x = 4\) which makes the expression positive.

Because the inequality is \( \leq 0\), we are interested in intervals where the test showed us the expression is negative or zero. Therefore, the solution includes intervals \((-\infty, 1] \) fully including 1, and \([2, 3]\) fully including both 2 and 3.