Problem 38 - Algebra 2

Question

Solve each inequality. Then graph the solution set on a number line. \(9 z+2>4 z+15\)

Step-by-Step Solution

Answer

The solution is \(z > \frac{13}{5}\); graph with an open circle at \(\frac{13}{5}\), shade to the right.

Step 1: Isolate Variable Terms

To solve the inequality, we first need to get all terms containing the variable "z" on one side. Begin by subtracting \(4z\) from both sides of the inequality.\[9z + 2 - 4z > 4z + 15 - 4z\]This simplifies to:\[5z + 2 > 15\]

Step 2: Eliminate Constant Terms

Next, isolate the term with the variable by removing the constant on the left side of the inequality. Subtract 2 from both sides of the inequality.\[5z + 2 - 2 > 15 - 2\]Which simplifies to:\[5z > 13\]

Step 3: Solve for the Variable

To find the value of "z", divide both sides of the inequality by 5.\[\frac{5z}{5} > \frac{13}{5}\]Thus,\[z > \frac{13}{5}\]

Step 4: Graph the Solution on a Number Line

Plot the solution \(z > \frac{13}{5}\) on the number line. Since \(z\) is greater than \(\frac{13}{5}\), draw an open circle at \(\frac{13}{5}\) and shade the line to the right of this point to indicate that all numbers greater than \(\frac{13}{5}\) satisfy the inequality.

Key Concepts

Inequality Solution

Inequalities are like equations, but instead of using an equal sign, they use inequality symbols such as '>','<', '≥', or '≤'. In the exercise provided, our goal was to solve the inequality, which means finding the set of all possible values for the variable "z" that satisfies the inequality.To solve the inequality given by \(9z + 2 > 4z + 15\),we need to "balance" the inequality, meaning whatever operation we perform on one side, we must perform on the other side as well. This keeps the inequality true. Through the steps of subtraction and division, we determined that\[z > \frac{13}{5}\].This result is the solution to the inequality, which states that any number greater than \(\frac{13}{5}\) will satisfy the inequality conditions established by the problem.

Number Line Graphing

Graphing the solution on a number line helps us visualize the range of solutions possible for the inequality. In the context of an inequality like\[z > \frac{13}{5}\],graphing involves placing points on a line to show the region of numbers that the inequality encompasses.Here’s how you can graph it:

• Identify the number \(\frac{13}{5}\) on the number line.
• At this position, draw an open circle. An open circle signifies that \(\frac{13}{5}\) itself is not included in the solutions, as we're dealing with a 'greater than' inequality.
• Shade the line extending to the right of \(\frac{13}{5}\), indicating all numbers greater than \(\frac{13}{5}\) are solutions.

This graphical representation helps understand how solution sets for inequalities differ from those of equations, which have exact values rather than ranges.

Isolation of Variable Steps

Isolating the variable is a critical step in solving both equations and inequalities. It involves manipulating the inequality so that the variable in question stands alone on one side, simplifying the process of identifying its potential values.In this exercise, the goal was to isolate "z" by first moving all terms with "z" to one side of the inequality. We achieved this step by subtracting a term:\[9z + 2 - 4z > 4z + 15 - 4z\],which simplifies to\[5z + 2 > 15\].This step sets the stage for removing constant terms and ultimately solving for the variable itself by isolating it.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions and equations or inequalities. This skill is crucial when solving any algebraic statement, whether it's an equation or inequality.For this inequality, algebraic manipulation involved:

• Subtracting \(4z\) from both sides to group variable terms together.
• Eliminating constants by subtracting \(2\) from both sides, isolating the variable term.
• Finally dividing through both sides by \(5\), scaling down the coefficient on "z" to isolate it completely.

Each of these steps requires balancing actions on both sides of the inequality. Remember, the key in algebraic manipulation is to perform operations that simplify and isolate the variable without altering the inequality’s solution set.