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Chapter 2: Problem 49

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

Short Answer

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

Step by step solution

01

Simplify the Inequality

Start by moving all terms involving the variable to one side of the inequality. Begin with subtracting \(4x\) from both sides of the inequality: \(6x - 2 - 4x > 4x - 14 - 4x\). This simplifies to \(2x - 2 > -14\).
02

Isolate the Variable

To isolate \(x\), add 2 to both sides of the inequality: \(2x - 2 + 2 > -14 + 2\). This results in \(2x > -12\).
03

Solve for x

Divide both sides by 2 to solve for \(x\): \(\frac{2x}{2} > \frac{-12}{2}\). This simplifies to \(x > -6\).
04

Express the Solution in Interval Notation

The inequality \(x > -6\) corresponds to all real numbers greater than -6. In interval notation, this is expressed as \((-6, \infty)\).

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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 way to succinctly describe a range of values, often used in mathematics, especially when dealing with inequalities. It uses brackets and parentheses to show closed and open ends. To write inequalities in interval notation:
  • If the interval is open, meaning it doesn鈥檛 include the endpoint, use parentheses \( ( ... , ... ) \).
  • If the interval is closed, meaning it includes the endpoint, use brackets \( [ ... , ... ] \).
In the solution \(x > -6\), the values of \(-6\) are not included, but all values greater than \(-6\) are. Thus, we use \( ( -6, \infty ) \). Here, the infinity symbol always pairs with a parenthesis because it's a concept, not a number. Infinity can never be an endpoint and is always open.
Variable Isolation
Isolating a variable is a key skill in solving equations and inequalities. It involves manipulating the equation or inequality so that the variable is by itself on one side. This often requires performing operations such as addition, subtraction, multiplication, or division to both sides of the inequality.
In the example, after simplifying the inequality to \(2x - 2 > -14\), the goal is to get \(x\) alone on one side. We achieve this in two steps:
  • First, add \(2\) to both sides to cancel the \(-2\) term next to \(x\), leading to \(2x > -12\).
  • Then, divide each side by \(2\) to isolate \(x\), resulting in \(x > -6\).
Always remember: whatever operation you do to one side of an inequality, do to the other side as well. This keeps the inequality balanced.
Algebraic Manipulation
Algebraic manipulation involves using algebra rules to rearrange and simplify expressions. This is crucial for dealing with inequalities successfully. These techniques include distributing, combining like terms, and reordering equations or inequalities.
Starting with an inequality like \(6x - 2 > 4x - 14\), you aim to simplify by bringing all variable terms to one side and constants to the other. Here's how:
  • Move terms: Start by subtracting \(4x\) from both sides to have all \(x\) terms on one side, simplifying to \(2x - 2 > -14\).
  • Combine like terms: This means getting all the \(x\) terms together and all the constant terms together, which makes the inequality simpler to solve.
Using these principles carefully ensures you're correctly simplifying and handling inequalities, leading to the precise solution.

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Most popular questions from this chapter

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -4 \leq \frac{x-1}{3} \leq 4 $$

For Problems \(35-44\), solve each compound inequality and graph the solution sets. Express the solution sets in interval notation. $$ 5 x-2<-2 \text { or } 5 x-2>2 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -25 \leq 4 x+3 \leq 19 $$

For Problems 45-56, solve each compound inequality using the compact form. Express the solution sets in interval notation. $$ -3<2 x+1<5 $$

Candace had scores of \(95,82,93\), and 84 on her first four exams of the semester. What score must she obtain on the fifth exam to have an average of 90 or better for the five exams?
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