Problem 14 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Write each solution set in interval notation. $$-3 x-8 \leq 7$$

Step-by-Step Solution

Answer

The solution is

Step 1: Isolate the x term

To isolate the term with the variable, add 8 to both sides of the inequality:

Step 2: Simplify the inequality

After adding 8 to both sides, the inequality becomes:

Step 3: Divide by -3

Now, divide both sides of the inequality by -3. Remember, dividing by a negative number reverses the inequality sign:

Step 4: Write in interval notation

The solution in interval notation is:

Key Concepts

Isolation of the Inequality

To solve an inequality, you first need to isolate the variable (in this case, let's focus on isolating \(x\)). Let's look at the inequality:
\[-3x - 8 \leq 7\]
Start by eliminating any constants on the side with the variable. Here, we'll add 8 to both sides:
\[-3x - 8 + 8 \leq 7 + 8\]
This simplifies to:
\[-3x \leq 15\]
Now, \(x\) is isolated with a coefficient of -3.

Reverse Inequality Sign

Handling inequalities involves a crucial rule: when you multiply or divide both sides of the inequality by a negative number, you must reverse the inequality sign. After isolating the term with \(x\), our inequality is:
\[-3x \leq 15\]
As the next step, we divide both sides by -3. Here's the key part—because we are dividing by a negative number, the inequality sign will flip:
\[x \geq -5\]
This step is essential for correctly solving inequalities. Always remember to reverse the inequality sign when dealing with negative multipliers or divisors.

Interval Notation

The final part of solving an inequality involves writing the solution in interval notation. This is a concise way to represent all possible solutions. From our final inequality \(x \geq -5\), it means \(x\) can be any number greater than or equal to -5.
In interval notation, this is written as:
\[[-5, \infty)\]
Here’s how to break it down:

• The bracket [ means -5 is included in the solution (\(x\) can be equal to -5).
• The parenthesis ) after infinity means infinity is not a specific number we can reach, so we use ( for open intervals.

This notation compactly expresses that \(x\) can be any number from -5 to infinity.