Problem 31 - College Algebra

Question

Solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation. $$ 2 x-5<-11 \text { or } 5 x+1 \geq 6 $$

Step-by-Step Solution

Answer

The solution is \( x < -3 \) or \( x \geq 1 \); in interval notation: \( (-\infty, -3) \cup [1, \infty) \).

Step 1: Solve the First Inequality

First, we'll address the inequality \( 2x - 5 < -11 \). To isolate \( x \), add 5 to both sides of the inequality: \( 2x - 5 + 5 < -11 + 5 \), which simplifies to \( 2x < -6 \). Next, divide both sides by 2 to solve for \( x \): \( \frac{2x}{2} < \frac{-6}{2} \) results in \( x < -3 \).

Step 2: Solve the Second Inequality

Now, solve the inequality \( 5x + 1 \geq 6 \). Start by subtracting 1 from both sides: \( 5x + 1 - 1 \geq 6 - 1 \), which simplifies to \( 5x \geq 5 \). Then, divide both sides by 5 to isolate \( x \), resulting in \( x \geq 1 \).

Step 3: Combine Inequality Solutions

The problem specifies an 'or' compound inequality, meaning we combine solutions where either inequality is satisfied. From Step 1, we have \( x < -3 \) and from Step 2, \( x \geq 1 \). Therefore, the solution is \( x < -3 \) or \( x \geq 1 \).

Step 4: Express using Interval Notation

Express the solution \( x < -3 \) or \( x \geq 1 \) in interval notation. The interval for \( x < -3 \) is \( (-\infty, -3) \) and for \( x \geq 1 \) is \( [1, \infty) \). Hence, the interval notation is \( (-\infty, -3) \cup [1, \infty) \).

Key Concepts

Understanding Inequality Signs

Inequality signs are symbols used to compare two values. They show us if one side is greater, less, equal, less than, or equal to the other. The key inequality signs you will encounter are:

• ">" (greater than)
• "<" (less than)
• ">=" (greater than or equal to)
• "<=" (less than or equal to)

In our exercise, we had two compound inequalities: \(2x - 5 < -11\) and \(5x + 1 \geq 6\). The sign "<" indicates that the left side must be smaller than the right, while "\geq" means the left side is either greater than or equal to the right.
These inequalities guide us in determining the range of values that \(x\) can take.

Mastering Interval Notation

Interval notation is a way to represent a set of numbers between two endpoints. It’s simple and efficient for expressing the solutions to inequalities.
Here are the basics:

• Parentheses "( )" are used when a number is not included.
• Brackets "[ ]" indicate the number is included.
• Infinity (\(\infty\) or \(-\infty\)) is always written with a parenthesis because it's not a specific number.

In our solution, \(x < -3\) translates to the interval \((-\infty, -3)\) because \(-3\) is not reached.
For \(x \geq 1\), we use \([1, \infty)\) since 1 is included based on the "\geq" sign.
The intervals are combined with a union symbol (\(\cup\)), leading to the final answer: \((-\infty, -3) \cup [1, \infty)\).
This notation clearly shows all solutions fulfilling either condition.

Solving Inequalities Step-by-Step

To solve compound inequalities like \(2x - 5 < -11\) or \(5x + 1 \geq 6\), we break it into manageable steps:

Step 1: Solve Each Inequality Separately

First, tackle each part of the inequality separately. Add or subtract terms to isolate the variable, and then divide or multiply to solve for \(x\).
For the first inequality, adding 5 to both sides and dividing by 2 gives \(x < -3\).
In the second inequality, subtract 1 and divide by 5, resulting in \(x \geq 1\).

Step 2: Combine the Solutions

The "or" in compound inequalities means either condition can be true.
We combine the solutions \(x < -3\) and \(x \geq 1\) because either one can satisfy the original condition.

Step 3: Express in Interval Notation

Finally, convert the solutions to interval notation:
- \((-\infty, -3)\) for \(x < -3\)
- \([1, \infty)\) for \(x \geq 1\)
These are put together using the union sign "\(\cup\)", showing all possible solutions in a neat format.