Problem 21 - Thomas Calculus

Question

Solve the inequalities in Exercises \(19-34,\) expressing the solution sets as intervals or unions of intervals. Also, show each solution set on the real line. $$ |t-1| \leq 3 $$

Step-by-Step Solution

Answer

The solution is \([-2, 4]\).

Step 1: Understanding the Absolute Value Inequality

The given inequality is \(|t - 1| \leq 3\). This can be interpreted as the distance between \(t\) and 1 on the number line being less than or equal to 3. This inequality includes both sides of the inequality \(-3 \leq t - 1 \leq 3\).

Step 2: Breaking Down the Compound Inequality

The inequality \(-3 \leq t - 1 \leq 3\) consists of two simpler inequalities. These are1. \(t - 1 \geq -3\)2. \(t - 1 \leq 3\)

Step 3: Solving the First Inequality

Solve \(t - 1 \geq -3\) by adding 1 to both sides, yielding \(t \geq -2\).

Step 4: Solving the Second Inequality

Solve \(t - 1 \leq 3\) by adding 1 to both sides, resulting in \(t \leq 4\).

Step 5: Combining the Solutions

The solutions from the previous steps are \(t \geq -2\) and \(t \leq 4\). Combining these, we get the inequality \(-2 \leq t \leq 4\), which represents the solution set.

Step 6: Expressing the Solution as an Interval

The solution \(-2 \leq t \leq 4\) can be expressed as the interval \([-2, 4]\).

Step 7: Visualizing on the Real Line

On the real line, the solution set \([-2, 4]\) is represented by a line segment that starts at -2 and ends at 4, including both endpoints because the inequalities are non-strict (\(\leq\)).

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities, like \(|t - 1| \leq 3\), represent the distance of a number from a specific point on the number line. In this case, it is the distance between \(t\) and 1 that must be less than or equal to 3. The absolute value \(|x|\) is essentially the same as saying "the distance of \(x\) from zero," and it is always non-negative.

When working with an absolute value inequality, it's useful to remember that \(|x| \leq a\) implies the compound inequality \(-a \leq x \leq a\). This means our original inequality \(|t - 1| \leq 3\) translates into \(-3 \leq t - 1 \leq 3\). The middle section here is your variable expression, bounded by negative and positive values of the absolute component.

Solving Compound Inequalities

Compound inequalities are a combination of two or more inequalities joined by 'and' or 'or'. They can represent a range of values that satisfy all parts of the inequality at once. For the expression \(-3 \leq t - 1 \leq 3\), you independently solve each side.

First, solve \(t - 1 \geq -3\) by adding 1 to both sides, which results in \(t \geq -2\). Then solve \(t - 1 \leq 3\) by similarly adding 1 to both sides, leading to \(t \leq 4\). Since they stem from the same inequality with conditions needing to hold true at the same time, the solutions need to overlap; in this context, it gives \( -2 \leq t \leq 4 \), a classic "and" compound inequality.

Thus, the solution set for our inequality reflects a continuous set of values that \(t\) can take.

Visual Representation on the Real Line

To represent inequalities on the real line, you identify the starting and ending points of your solution set. For \(-2 \leq t \leq 4\), you’ll mark a solid dot at \(-2\) to indicate that it’s included in the solution. Do the same with point 4.

Then, draw a line connecting these two points. This line segment, inclusive of the endpoints, visually signifies all the possible values \(t\) can assume. The key here is understanding that the line represents the range specified by the inequality and helps in visualizing the solution.

Expressing Solutions with Interval Notation

Interval notation is a shorthand used to represent the set of solutions for inequalities. It uses brackets to denote endpoints. Square brackets \([a, b]\) indicate that both \(a\) and \(b\) are included in the solution set, suitable for non-strict inequalities like \(\leq\) or \(\geq\).

For our solution \(-2 \leq t \leq 4\), it is expressed in interval notation as \([-2, 4]\). This compact form of notation quickly communicates the range of numbers included in the solution. Understanding this notation is crucial for easily reading and writing solutions.