Problem 41 - Algebra for College Students

Question

For Problems \(41-70\), solve each inequality and express the solution set using interval notation. 2 x-1>6

Step-by-Step Solution

Answer

\( (\frac{7}{2}, \infty) \)

Step 1: Isolating the Variable

Start by adding 1 to both sides of the inequality:\[ 2x - 1 + 1 > 6 + 1 \]Simplifying the equation gives:\[ 2x > 7 \]

Step 2: Solving for x

Now, divide both sides of the inequality by 2 to solve for \( x \):\[ \frac{2x}{2} > \frac{7}{2} \]This simplifies to:\[ x > \frac{7}{2} \]

Step 3: Expressing the Solution in Interval Notation

The inequality \( x > \frac{7}{2} \) means that \( x \) is greater than \( \frac{7}{2} \), but does not include \( \frac{7}{2} \) itself. Thus, the solution in interval notation is:\[ \left( \frac{7}{2}, \infty \right) \]

Key Concepts

Interval Notation

Interval notation is a way to represent a range of numbers, commonly used to describe the solution set of an inequality.
It is compact and straightforward.

• In interval notation, parentheses \((\) and \()\) are used to show that an endpoint is not included in the interval. For instance, \((a, b)\) means all numbers greater than \(a\) and less than \(b\), but not the values \(a\) or \(b\) themselves.
• If an endpoint is included in the interval, square brackets \([\) and \()]\) are used. For example, \([a, b]\) includes both \(a\) and \(b\).
• Intervals can extend indefinitely. For instance, \((a, \infty)\) signifies all numbers greater than \(a\), and \(-\infty, b)\) includes all numbers less than \(b\).

In our exercise, we used interval notation \(\left( \frac{7}{2}, \infty \right)\) to express that \(x\) is greater than \(\frac{7}{2}\) but not equal to it, stretching indefinitely to the larger numbers.

Isolation of Variable

The process of isolating the variable is about rearranging an equation or inequality so that the variable appears by itself on one side.
This step is essential when solving inequalities.To isolate the variable properly:

• First, perform operations on both sides of the inequality that simplify the expression around the variable. For example, in the inequality \(2x - 1 > 6\), we added 1 to both sides to eliminate the constant term on the side with the variable.
• Next, divide or multiply to make the coefficient of the variable equal to 1. Here, dividing both sides by 2 allowed us to solve for \(x\), giving \(x > \frac{7}{2}\).
• Always reverse the inequality sign when multiplying or dividing by a negative number. Since we did not encounter a negative division or multiplication in this exercise, the direction of the inequality stayed the same.

By understanding how to effectively isolate the variable, solving inequalities becomes a systematic process.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations.
They form the basis for creating and solving mathematical problems.Subtopics in algebraic expressions include:

• Terms: Parts of the expression separated by addition or subtraction signs. In \(2x - 1\), \(2x\) and \(-1\) are terms.
• Coefficients: Numbers in front of variables. Here, \(2\) is the coefficient of \(x\).
• Constants: Numbers without variables, like \(-1\) in our expression.
• Operations: Determine the transformation of expressions, such as addition, subtraction, multiplication, and division.

Understanding algebraic expressions allows you to manipulate equations and inequalities to reach a solution. Each step, like simplifying by combining like terms or balancing an equation, relies on a firm grasp of these concepts.