Problem 53 - College Algebra

Question

Solve compound inequality. \(-3 \leq x-2 \leq 1\)

Step-by-Step Solution

Answer

The solution to the compound inequality \(-3 \leq x-2 \leq 1\) is \(-1 \leq x \leq 3\).

Step 1: Address the left inequality

Look at the left half of the inequality first: \(-3 \leq x-2\). Isolate \(x\) by adding 2 to both sides: \(-3 + 2 \leq x-2 + 2\), which simplifies to \(-1 \leq x\).

Step 2: Address the right inequality

Next, look at the right half of the inequality: \(x-2 \leq 1\). Isolate \(x\) by adding 2 to both sides: \(x-2 + 2 \leq 1 + 2\), which simplifies to \(x \leq 3\).

Step 3: Combine the two inequalities

We've found \(x\) is greater than or equal to -1 and less than or equal to 3. This can be written as a compound inequality: \(-1 \leq x \leq 3\). This means any value of \(x\) in the closed interval [-1,3] is a solution.

Key Concepts

Understanding Compound Inequalities

A compound inequality is a combination of two or more inequalities that are connected by the words 'and' or 'or.' When working with compound inequalities where both conditions must be true, we are dealing with an 'and' situation. In our exercise, \(-3 \leq x-2 \leq 1\), there are two inequalities joined by 'and,' which means we need to find the values of \(x\) that satisfy both inequalities simultaneously.

By addressing each part of the compound inequality separately, like in the provided steps, and then combining them, we ensure we capture the full range of values for \(x\) that make the inequality true. The solution to the compound inequality tells us the set of all possible values of \(x\) that satisfy both conditions at the same time. A key takeaway is that the intersection or overlap of the solutions to each individual inequality gives us the final solution to a compound inequality with 'and.'

Deciphering Inequality Notation

Inequality notation is used to express a range of values that a variable can take. In our exercise example, we're looking at \(-3 \leq x-2 \leq 1\), where \(-3 \leq x-2\) and \(x-2 \leq 1\) indicate the lower and upper bounds of \(x\), respectively. The symbols \(\leq\) and \(\geq\) stand for 'less than or equal to' and 'greater than or equal to,' while \(<\) and \(>\) are 'less than' and 'greater than', without including the number itself.

In the step-by-step solution, we add 2 to all parts of the inequality to isolate \(x\), which gives us the simplified form \(-1 \leq x \leq 3\). It's important to follow the correct order of operations and properly apply algebraic principles when solving inequalities, just as one would with equations.

Interval Notation and Its Use in Inequalities

Interval notation gives us a convenient way to represent ranges of values on the number line. It takes the format of two numbers, representing the lower and upper bounds, enclosed in brackets or parentheses. Square brackets, like \([a, b]\), indicate that the endpoints a and b are included in the interval—we call this a closed interval. Parentheses, like \((a, b)\), mean that a and b are not included—this is an open interval.

In the provided solution, the compound inequality \(-1 \leq x \leq 3\) translates to the closed interval \([-1,3]\) in interval notation. This means all the numbers from -1 to 3, including -1 and 3 themselves, are solutions to the inequality. Understanding interval notation is essential for succinctly expressing the solution sets of inequalities.