Problem 65 - Elementary Algebra

Question

Solve and graph the solution set. In addition, present the solution set in interval notation. $$ -3 \leq 3(x-1) \leq 3 $$

Step-by-Step Solution

Answer

Solution: \([0, 2]\); graph: line from 0 to 2 with filled circles on each.

Step 1: Understanding the Compound Inequality

The given inequality is \(-3 \leq 3(x-1) \leq 3\). This is a compound inequality that needs to be broken into two separate inequalities to make it easier to solve: \(-3 \leq 3(x-1)\) and \(3(x-1) \leq 3\).

Step 2: Solve the First Inequality

First, solve \(-3 \leq 3(x-1)\). Divide both sides by 3 to get \(-1 \leq x-1\). Then, add 1 to both sides to obtain \(0 \leq x\) or \(x \geq 0\).

Step 3: Solve the Second Inequality

Now, solve \(3(x-1) \leq 3\). Divide both sides by 3 to get \(x-1 \leq 1\). Then, add 1 to both sides to obtain \(x \leq 2\).

Step 4: Combine Solutions

Combine the solutions from steps 2 and 3: \(0 \leq x \leq 2\). This means \(x\) is greater than or equal to 0 and less than or equal to 2.

Step 5: Express Solution in Interval Notation

The solution in interval notation is \([0, 2]\).

Step 6: Graph the Solution

To graph the solution on a number line, draw a line from 0 to 2, including both endpoints with filled circles to indicate that 0 and 2 are part of the solution set.

Key Concepts

solving inequalities

When solving inequalities, it's essential to break down complex expressions into simpler parts. This involves cases like compound inequalities, where you have expressions connected by two comparison operators, such as "\(-3 \leq 3(x-1) \leq 3\)." Here, we break it into two separate inequalities to manage them easier:

• \(-3 \leq 3(x-1)\)
• \(3(x-1) \leq 3\)

Each inequality needs to be solved independently:

• For \(-3 \leq 3(x-1)\), simplify by dividing each side by 3, leading to \(-1 \leq x-1\). Adding 1 to each side gives you \(0 \leq x\) or equivalently, \(x \geq 0\).
• For \(3(x-1) \leq 3\), divide each side by 3 to get \(x-1 \leq 1\). Adding 1 results in \(x \leq 2\).

These two solutions combine to form a range of values that satisfy both conditions simultaneously: \(0 \leq x \leq 2\). This means \(x\) can be any number from 0 to 2, inclusive.

graph solution

Graphing the solution of an inequality helps visualize the range of possible values for \(x\). With our compound inequality solution \(0 \leq x \leq 2\), the graph on a number line is straightforward. Here’s how to do it step by step:

• Start by drawing a number line with a reasonable scale covering at least the points from 0 to 2.
• Mark the points 0 and 2 on the line. Since our solution includes these boundary values – indicated by the \(\leq\) (less than or equal to) symbol – use filled circles to denote they are part of the solution set.
• Draw a solid line connecting the points 0 and 2, representing all numbers \(x\) that fall within this interval.

This visual representation confirms that \(x\) can be any value starting from 0 up to and including 2.

interval notation

Interval notation provides a concise way to represent the set of solutions for inequalities. It uses interval brackets to denote the continuous range of values. For our solution \(0 \leq x \leq 2\), the interval notation expresses this range simply as \([0, 2]\).
Here's what the components signify:

• The square bracket \([\) next to 0 indicates that the interval includes 0. This is consistent with the \(\geq\) or \(\leq\) inequalities in mathematics, meaning the endpoint is part of the solution.
• The square bracket next to 2 also indicates that 2 is included in the set. The solution is inclusive of its boundary numbers.

This form neatly summarizes the range \(x\) can occupy and is particularly useful for communicating solutions quickly and accurately in mathematical discussions and written work.