Problem 60 - Intermediate Algebra

Question

Solve each inequality. Graph the solution set and write it using interval notation. $$ 5(3+z)>-3(z+3) $$

Step-by-Step Solution

Answer

The solution is \( (-3, +\infty) \).

Step 1: Expand Both Sides

Start by expanding both sides of the inequality. Distribute the 5 on the left-hand side and the -3 on the right-hand side:On the left: \[ 5(3 + z) = 15 + 5z \]On the right:\[ -3(z + 3) = -3z - 9 \]

Step 2: Combine the Inequality

Substitute the expanded terms back into the inequality:\[ 15 + 5z > -3z - 9 \]

Step 3: Move Variable Terms to One Side

Add \(3z\) to both sides to get all variable terms on one side:\[ 15 + 5z + 3z > -9 \]This simplifies to:\[ 15 + 8z > -9 \]

Step 4: Isolate the Variable Term

Subtract 15 from both sides to isolate the term with \(z\):\[ 8z > -9 - 15 \]This simplifies to:\[ 8z > -24 \]

Step 5: Solve for \(z\)

Divide both sides by 8 to solve for \(z\):\[ z > \frac{-24}{8} \]Which simplifies to:\[ z > -3 \]

Step 6: Write the Solution in Interval Notation

The solution set for \(z > -3\) in interval notation is:\((-3, +\infty)\)

Step 7: Graph the Solution Set

On a number line, draw an open circle at \(-3\) and shade the line extending to the right, indicating all values greater than \(-3\).

Key Concepts

Interval Notation

Interval notation is a method used to describe a set of numbers or a solution set for inequalities in a compact and precise way. In our example, we arrived at the solution for the inequality as \(z > -3\). This means that \(z\) can take any value greater than \(-3\). We express this with interval notation as \((-3, \infty)\).

• The parentheses \(()\) are used because \(-3\) is not included in the solution—it's an open boundary.
• The infinity symbol \(\infty\) always has a parenthesis because infinity isn't a specific number we can reach.

This concise notation tells us that the solution set extends infinitely in one direction. It’s perfect when you want to express solutions quickly without drawing a graph every time.

Graphing Inequalities

Graphing inequalities involves representing the solution set of an inequality visually on a number line. For \(z > -3\), we want to show all possible values greater than \(-3\).

• First, identify the number that your variable is greater than—in this case, \(-3\).
• Place an open circle on the number line at \(-3\) to indicate that \(-3\) itself is not included.
• Shade the region to the right of the \(-3\) to show where all numbers greater than \(-3\) lie.

This visual representation helps you and others see at a glance where the solution set lies, making it easier to understand than just reading the inequality.

Solving Linear Inequalities

Solving linear inequalities involves similar steps to solving equations, with a few additional considerations. For example, when we solved the inequality \(5(3+z)>-3(z+3)\), we used several steps:

• Expand each side: Distribute the numbers outside the parentheses to get rid of them so you have a clear algebraic expression to work with.
• Combine like terms: Rearrange the inequality to have all terms with the variable on one side.
• Isolate the variable: Move constants to the other side to make one side only about the variable.
• Solving: Divide or multiply to solve for the variable, remembering if you multiply or divide by a negative number, flip the inequality sign.

The key here is maintaining a balance, just like with equations, but remembering the rule about flipping the inequality sign, which is unique to inequalities. After reaching a solution for \(z > -3\), it’s important to express it in both interval notation and graphically, ensuring full comprehension of the solution set.