Problem 39 - Beginning and Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write the answer in interval notation. $$|z-3| \geq-5$$

Step-by-Step Solution

Answer

The solution to the inequality \(|z-3| \geq -5\) is \(-2 \leq z \leq 8\). Graphically, this is a solid line between (and including) the points z = -2 and z = 8 on the number line. In interval notation, this is written as \([-2, 8]\).

Step 1: Understand the Problem

We are given an inequality with an absolute value: \(|z-3| \geq -5\). The goal is to solve this inequality for z, then graph the solution set, and finally write the answer in interval notation.

Step 2: Deal with the absolute value

Recall that the definition of the absolute value is: \(|x| = x\) if \(x \geq 0\) and \(|x| = -x\) if \(x < 0\). We can apply this to our inequality to break it into two separate cases. Case 1: When \(z-3\) is positive or zero (\(z - 3 \geq 0\)): \( |z-3| = z - 3\) In this case, our inequality becomes: \(z-3 \geq -5\) Case 2: When \(z-3\) is negative (\(z-3 < 0\)): \( |z-3| = -(z-3)\) In this case, our inequality becomes: \(-(z-3) \geq -5\)

Step 3: Solve the two separate inequalities

Now let's solve each inequality separately: Case 1: \(z - 3 \geq -5\) Add 3 to both sides: \(z \geq -2\) Case 2: \(-(z-3) \geq -5\) Expand the inequality: \(-z + 3 \geq -5\) Add z to both sides: \(3 \geq -5 + z\) Add 5 to both sides: \(8 \geq z\)

Step 4: Combine the solutions

We found that \(z \geq -2\) and \(8 \geq z\) when considering both cases. To get the overall solution, we must take the intersection of both solutions. This means that we want all z values that satisfy both inequalities, which is given by: \(-2 \leq z \leq 8\)

Step 5: Graph the solution set

To graph the solution set, draw a number line and mark the interval between -2 and 8, including the endpoints. This should look like a closed circle at z = -2 and z = 8 connected by a solid line in between.

Step 6: Write the answer in interval notation

In interval notation, the solution is written as a closed interval since both endpoints are included in the solution set. The final answer in interval notation is: \([-2, 8]\)

Key Concepts

Absolute Value Inequalities

Absolute value inequalities help us determine relationships about the magnitude of numbers. The main idea is to measure the distance a number is from zero on a number line. For instance, the absolute value of \(|x|\) symbolizes how far x is from zero, no matter if x is positive or negative.
When handling inequalities involving absolute values, the process slightly changes:

• If we have \(|x| \geq c\) (where c is a positive number), it means x is at least c units away from zero. This breaks into two cases: \(x \geq c\) and \(x \leq -c\).
• If we have \(|x| \leq c\), it means x is at most c units away from zero, leading to \(-c \leq x \leq c\).

Understanding these concepts makes solving inequalities with absolute values a breeze!

Graphing Solution Sets

Graphing solution sets is a visual technique that clarifies the set of solutions to an equation or inequality on a number line. It offers a clear picture of the solution's range and limits.
When graphing, remember:

• Closed circles indicate the endpoint is included in the solution (like \(\leq\) or \(\geq\) ).
• Open circles show the endpoint isn't part of the solution ( \(<\) or \(>\) ).

For our inequality \(-2 \leq z \leq 8\), plot a closed circle at -2 and another at 8, then shade the area in between. This accurately represents all possible z values satisfying the inequality.

Interval Notation

Interval notation is a succinct way to describe sets of numbers that lie between boundaries. It’s widely used because of its simplicity and precision. Here's how to read and write interval notations:

• Brackets \([, ]\) are used when the endpoints are included, known as closed intervals.
• Parentheses \((, )\) indicate that the endpoints aren't included, known as open intervals.

For instance, the interval \([-2, 8]\) includes all real numbers from -2 to 8, with both -2 and 8 included. Using this notation makes complex ideas simpler and more accessible.

Algebraic Concepts

Algebra is the backbone of solving equations and inequalities. It involves using operations to isolate variables, allowing us to find their possible values. Key principles include:

• Addition and subtraction: You can add or subtract the same value from both sides of the equation without changing the equation's balance.
• Multiplication and division: You can multiply or divide each side by the same non-zero number.

Applying these principles to our inequalities, you might start by rearranging terms or expanding expressions. This step-by-step process untangles complex inequalities and creates a path to the solution. Understanding these basics makes navigating through algebraic problems much smoother.