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Chapter 2: Problem 41

Solve each inequality and express the solution set using interval notation. \(2 x-1>6\)

Short Answer

Expert verified
Solution set: \((\frac{7}{2}, \infty)\).

Step by step solution

01

Isolate the Variable Term

First, start by getting the term with the variable alone on one side of the inequality. We have:\[2x - 1 > 6\]Add 1 to both sides to remove the constant term from the left side:\[2x - 1 + 1 > 6 + 1\]This simplifies to:\[2x > 7\]
02

Solve for the Variable

Now, solve for \(x\) by dividing both sides of the inequality by 2. This gives:\[\frac{2x}{2} > \frac{7}{2}\]Simplifying, we find:\[x > \frac{7}{2}\]
03

Express the Solution in Interval Notation

The inequality \(x > \frac{7}{2}\) can be expressed in interval notation as \(\left( \frac{7}{2}, \infty \right)\). This means \(x\) can be any number greater than \(\frac{7}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Interval notation is a method used to represent the solution set of inequalities. It uses brackets and parentheses to express a range of values. For instance, when you have a solution like \(x > \frac{7}{2}\), it implies that \(x\) is any number greater than \(\frac{7}{2}\).
This is written as \((\frac{7}{2}, \infty)\) in interval notation.
  • Round brackets \(()\) indicate that the endpoint is not included, known as an "open" interval.
  • Square brackets \([]\) indicate the endpoint is included, known as a "closed" interval. This is used when there is a \(\geq\) or \(\leq\) sign.
The interval notation \((\frac{7}{2}, \infty)\) includes numbers starting from a bit more than \(\frac{7}{2}\) and goes up to positive infinity. Infinity always gets a round bracket because it is not a concrete number that can be reached.
Variable Isolation
Variable isolation is the first step in solving any inequality. The main aim is to get the variable, often denoted as \(x\), alone on one side of the inequality sign. This process involves reversing operations that are on the variable's side.To isolate a variable, you may need to:
  • Add or subtract numbers from both sides to eliminate constants.
  • Multiply or divide both sides by a number to undo any coefficients in front of the variable.
In the inequality \(2x - 1 > 6\), you start by removing the \(-1\) from the left side by adding 1 to both sides. This simplifies the expression to \(2x > 7\). Next, divide both sides by 2 to solve for \(x\), giving \(x > \frac{7}{2}\). Through these steps, \(x\) is isolated.
Inequality Solution Steps
Solving an inequality requires a systematic approach, similar to solving equations, but with special attention to the inequality sign.Here are the general steps for solving inequalities:
  • **Isolate the variable**: Start by getting the variable term on one side using addition or subtraction.
  • **Solve for the variable**: Then, get the variable completely alone by dividing or multiplying through by any coefficients.
For our problem of \(2x - 1 > 6\):1. Add 1 to both sides to neutralize the negative constant: \(2x - 1 + 1 > 6 + 1\) simplifies to \(2x > 7\).2. Next, divide both sides by 2: \(\frac{2x}{2} > \frac{7}{2}\), which simplifies to \(x > \frac{7}{2}\).
Keep in mind, if multiplying or dividing by a negative number, reverse the inequality sign. This is crucial for maintaining the inequality's truth.

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