Problem 79 - College Algebra with Modeling and Visualization

Question

Solve the inequality. Approximate the endpoints to the nearest thousandth when appropriate. $$ 2 x-8>5 $$

Step-by-Step Solution

Answer

The solution is \(x > 6.500\).

Step 1: Isolate the variable term

Start by getting the term with the variable, \(2x\), by itself. To do this, add 8 to both sides of the inequality to eliminate the -8 from the left side. You will have: \[ 2x > 5 + 8 \] This simplifies to: \[ 2x > 13 \]

Step 2: Solve for the variable

Now that \(2x\) is isolated, divide both sides of the inequality by 2 to solve for \(x\):\[ x > \frac{13}{2} \] This simplifies to: \[ x > 6.5 \]

Step 3: Interpret the solution

The inequality \(x > 6.5\) indicates that all values greater than 6.5 satisfy the original inequality. This is the solution in an approximate form, with the endpoint rounded to the nearest thousandth.

Key Concepts

Algebra

Algebra is a branch of mathematics that involves expressions with variables. Variables are symbols or letters that represent numbers and can change values. Algebra simplifies and solves equations or inequalities involving these variables. In this exercise, algebraic methods were used to manipulate the inequality. By systematically adding or dividing, we aim to find the value of the variable that satisfies the inequality. When solving algebraic problems, it's important to follow a logical order of operations. This typically involves:

• Step-by-step manipulation of expressions.
• Combining like terms.
• Solving for unknowns in a clear and structured manner.

Variable Isolation

Variable isolation is a crucial technique in algebra. It involves rearranging an equation or inequality so the variable of interest is on one side, making it easier to solve. For our inequality, the variable term was initially part of a larger expression. We had to manipulate the equation to leave the variable on its own. In this particular step:

• We eliminated any constants accompanying the variable by adding or subtracting them from both sides of the inequality.
• We then simplified all terms to focus solely on the variable term, simplifying our path towards solving it.

Understanding variable isolation helps us perceive inequations or equations more clearly. It lays the groundwork for deriving exact solutions or bounds.

Inequality Interpretation

Interpreting inequalities involves understanding what the mathematical expression means in terms of ranges of numbers. After isolating the variable and solving the inequality, we get a solution in the form of an inequality. In our solved example, the solution came out as:

• x > 6.5

This indicates that any value of x that is greater than 6.5 will satisfy the inequality. It is important to note that inequalities can describe intervals in a number line and often need to be approximated. Rounding to a specified decimal point gives a practical representation. Reading and interpreting inequalities requires recognizing boundaries. It also means understanding inequalities allow for range-based answers, not just specific numbers like equations typically do.