Problem 29 - College Algebra with Corequisite Support

Question

For the following exercises, solve the compound inequality. Express your answer using inequality signs, and then write your answer using interval notation. $$ 3 x+1>2 x-5>x-7 $$

Step-by-Step Solution

Answer

\(x > -2\); interval notation: \((-2, \infty)\).

Step 1 - Break the Compound Inequality

The given compound inequality is \(3x + 1 > 2x - 5 > x - 7\). We can break this into two separate inequalities:1. \(3x + 1 > 2x - 5\)2. \(2x - 5 > x - 7\).

Step 2 - Solve the First Inequality

For the inequality \(3x + 1 > 2x - 5\), subtract \(2x\) from both sides:\[3x + 1 - 2x > -5\]This simplifies to:\[x + 1 > -5\]Next, subtract 1 from both sides:\[x > -6\].

Step 3 - Solve the Second Inequality

For the inequality \(2x - 5 > x - 7\), subtract \(x\) from both sides:\[2x - x - 5 > -7\]This simplifies to:\[x - 5 > -7\]Add 5 to both sides to isolate \(x\):\[x > -2\].

Step 4 - Combine the Solutions

Combine the results from Step 2 and Step 3 to find the solution to the original compound inequality. We solve for:\[x > -6\] and \[x > -2\].Since both conditions must be true simultaneously, we take the more restrictive condition:\[x > -2\].

Step 5 - Express Solution with Interval Notation

The inequality \(x > -2\) can be expressed in interval notation as:\((-2, \infty)\). This represents all real numbers greater than \(-2\).

Key Concepts

Inequality Signs

Inequality signs are symbols used to compare two values or expressions. They show us whether one value is larger, smaller, or simply different from another value. In the world of compound inequalities, like the exercise you're working with, these signs help to communicate more complex relationships between numbers. The two most common inequality signs you'll see are the greater than ">" and less than "<" symbols.

When dealing with inequalities, it's crucial to understand how each sign works:

• Greater than ">": Indicates that the value on the left side is larger than the one on the right.
• Less than "<": Indicates that the value on the left side is smaller than the one on the right.
• Greater than or equal to "≥": Shows that the value on the left is either greater than or equal to the right.
• Less than or equal to "≤": Shows that the value on the left is either less than or equal to the right.

Compound inequalities combine two or more inequalities joined by the word "and" or "or". An "and" compound inequality means both conditions must be true at the same time. An "or" means at least one of the conditions needs to be true. Understanding these signs helps you correctly interpret and solve mathematical expressions!

Interval Notation

Interval notation is a method used to describe a range of numbers, making it easier to present the solution of inequalities in a concise, easy-to-read format. When solving inequalities, like we did in the exercise, the final solution can often be neatly expressed using interval notation.

The key components of interval notation include:

• Brackets "[ ]" are used to show that an endpoint is included in the interval, known as a closed interval.
• Parentheses "( )" are used to indicate that an endpoint is not included, known as an open interval.
• Infinity '∞' is used whenever the interval continues indefinitely in the positive direction, and negative infinity '-∞' for the opposite. Remember, infinity is always accompanied by parentheses since it's not a reachable point.

Let's look at the answer we derived: the inequality was expressed as \(x > -2\) and in interval notation, this is written as \((-2, \infty)\). It includes all numbers greater than \-2\, but not \(-2\) itself, hence the use of an open bracket next to \(-2\). Understanding interval notation is essential when communicating the solutions of inequalities, as it provides a universal way to represent number ranges.

Solving Inequalities

Solving inequalities is a fundamental skill in algebra, requiring a methodical approach to find solutions that satisfy the given conditions. Think of inequalities similarly to solving equations, but with some additional rules to keep in mind.

Follow these steps for solving:

• First, isolate the variable on one side of the inequality sign using inverse operations, such as addition, subtraction, multiplication, or division, just like you would when solving equations.
• Remember one crucial rule: when multiplying or dividing both sides by a negative number, flip the inequality sign.
• Check if the inequality is compound, splitting it into two simpler inequalities if necessary (as demonstrated in the original exercise).
• Solve each inequality individually, then combine the results.

Once you've solved the inequalities, you can express the solutions in various ways, including using inequality signs or interval notation. In our example, the solution of the compound inequality \(3x + 1 > 2x - 5 > x - 7\) resulted in \(x > -2\). This solution is sometimes more restrictive than initial inequalities due to the nature of compound conditions, helping us identify the most precise answer that fits all given parameters.