Problem 49 - Algebra for College Students

Question

Solve each inequality and express the solution set using interval notation. \(6 x-2>4 x-14\)

Step-by-Step Solution

Answer

The solution is \((-6, \infty)\).

Step 1: Eliminate like terms

To solve the inequality, let's first eliminate the similar terms on both sides. The inequality is given by: \[6x - 2 > 4x - 14\]We need to bring all terms with \(x\) to one side, so let's subtract \(4x\) from both sides:\[(6x - 4x) - 2 > 4x - 4x - 14\]This simplifies to:\[2x - 2 > -14\]

Step 2: Isolate the variable term

The goal is to isolate the term with \(x\) on one side. Start by getting rid of the \(-2\) on the left side by adding \(2\) to both sides:\[2x - 2 + 2 > -14 + 2\]Simplify this to:\[2x > -12\]

Step 3: Solve for the variable

Now we need to solve for \(x\) by dividing both sides by \(2\) to get:\[\frac{2x}{2} > \frac{-12}{2}\]This simplifies to:\[x > -6\]

Step 4: Express the solution in interval notation

The solution to the inequality \(x > -6\) can be expressed using interval notation. Since \(x\) is greater than \(-6\), the solution set is:\((-6, \infty)\)

Key Concepts

Interval Notation

Interval notation is a simple and concise way to express the range of solutions for inequalities. It uses brackets and parentheses to indicate the inclusion or exclusion of boundary numbers. For example, in the solution set for the inequality \(x > -6\), we use the notation \((-6, \infty)\). Here's a breakdown:

• Parentheses \(( \text{ and } )\) are used when the endpoint is not included in the set. This is known as an "+open interval+". In our example, \(-6\) is not part of the solution set because the inequality is strictly greater than, hence \((-6,\infty)\).
• Infinity \(\infty\) always uses parentheses because it represents all numbers beyond a certain point and cannot be included as an exact value.

Understanding this notation allows for quick visualization of the solution set, making it easier to comprehend the range of values that satisfy the inequality.

Isolate the Variable Term

Isolating the variable is a crucial step in solving inequalities. It involves manipulating the inequality to get the variable by itself on one side of the equation. This helps in clearly understanding what values the variable can take.
To isolate the variable in the inequality \(6x - 2 > 4x - 14\), we start by getting all terms containing \(x\) to one side. This often involves both adding and subtracting terms. Let's go through this step-by-step:

• First, we eliminate any \(x\)-terms from the right side by subtracting \(4x\) from both sides. This gives us: \(6x - 4x - 2 > -14\).
• Next, we simplify it to \(2x - 2 > -14\).
• Then, we move the constant term \(-2\) by adding \(2\) to both sides, further simplifying it to \(2x > -12\).

The focus is on getting the variable \(x\) all by itself. This makes the inequality easier to solve and interpret.

Eliminate Like Terms

Eliminating like terms is the process where we combine similar terms to simplify inequalities. When you encounter terms with the same variable, they can be straightforwardly combined to either side of the inequality. This is essential as it helps in reducing the expression for easier manipulation.
In our initial inequality \(6x - 2 > 4x - 14\), here's what happens:

• Identify like terms: Here, the terms \(6x\) and \(4x\) are both expressions of \(x\). Their coefficients can be subtracted directly.
• Subtract \(4x\) from both sides to consolidate the \(x\)-terms on the left:\(6x - 4x - 2 > -14\).
• After subtraction, we get \(2x - 2 > -14\), making the inequality simpler and ready for further steps.

This simplification eliminates unnecessary complexity, allowing for a more direct path to the solution. By removing redundant terms, the inequality becomes much clearer and manageable.

Expressing Solutions

Expressing solutions is about translating the result of an inequality into a format that succinctly communicates the set of numbers that satisfy the condition. After solving the inequality, the final solution needs to be expressed in a universally understandable way.
For \(x > -6\), the solution reads as all numbers greater than \(-6\). In mathematical terms, this is expressed using interval notation as \((-6,\infty)\).

• This notation effectively communicates that \(-6\) is not included in the solution. Only numbers greater than \(-6\) apply.
• Using the symbol \(\infty\) indicates that there is no upper boundary, and the solution extends indefinitely to larger numbers.

Expressing the set in this way not only simplifies communication but also aligns with mathematical conventions, aiding in broader understanding.