Problem 29 - College Algebra

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

The solution is \(x > -2\), which is \((-2, \infty)\) in interval notation.

Step 1: Break Down the Compound Inequality

The given compound inequality is \(3x + 1 > 2x - 5 > x - 7\). To solve this, we need to break this into two separate inequalities: 1. \(3x + 1 > 2x - 5\)2. \(2x - 5 > x - 7\).

Step 2: Solve the First Inequality (Part 1)

Let's solve \(3x + 1 > 2x - 5\):Subtract \(2x\) from both sides to get:\(3x - 2x + 1 > - 5\)Simplifying gives:\(x + 1 > -5\)Subtract 1 from both sides:\(x > -6\).

Step 3: Solve the Second Inequality (Part 2)

Now solve \(2x - 5 > x - 7\):Subtract \(x\) from both sides:\(2x - x - 5 > -7\)Which simplifies to:\(x - 5 > -7\)Add 5 to both sides:\(x > -2\).

Step 4: Combine the Solutions

Now, combine the results from both parts:From \(x > -6\) and \(x > -2\), we take the more restrictive condition.The solution is \(x > -2\).

Step 5: Express the Solution in Interval Notation

Since \(x > -2\), the interval notation for this inequality is \((-2, \infty)\).

Key Concepts

Inequality Signs

Inequality signs are symbols used to compare the sizes or values of two numbers or expressions. In mathematics, inequalities are often solved to find a range of values that satisfy a certain condition. There are four primary inequality signs used:

• Greater than (">"),
• Less than ("<"),
• Greater than or equal to ("≥"), and
• Less than or equal to ("≤").

When dealing with compound inequalities, as in the exercise, you're often comparing values across two or more different inequalities simultaneously. This means you'll solve each part separately before combining the results. In the case of strict inequalities like ">" or "<", the value at the boundary is not included. Remember to pay close attention to these symbols during calculations, as they determine which values satisfy the inequality.

Interval Notation

Interval notation is a shorthand method for describing ranges of values, particularly those that satisfy an inequality. This method uses brackets and parentheses to indicate whether endpoints are included or not in the interval. Here’s how you can understand it:

• Parentheses "(", ")" are used to denote that an endpoint is not included, meaning values are strictly inside the endpoint range.
• Brackets "[", "]" are used when an endpoint is included, implying that the boundary value is part of the solution set.

In the solution to our exercise, the compound inequality leads to a solution of "x > -2". This tells us that values greater than -2 satisfy our inequality, excluding -2 itself. Hence, the appropriate interval is the open interval (-2, ∞), which signifies all values greater than -2 up to positive infinity.

Algebraic Solutions

Finding algebraic solutions to compound inequalities involves step-by-step manipulation to isolate the variable of interest. In the exercise provided, we worked with the compound inequality \(3x + 1 > 2x - 5 > x - 7\). Here's how we tackled it:First, we broke it down into smaller, more manageable parts. This means dividing the compound inequality into separate inequalities like \(3x + 1 > 2x - 5\) and \(2x - 5 > x - 7\). Solving these involved basic algebraic steps:

• For \(3x + 1 > 2x - 5\), we simplified and rearranged to find \(x > -6\).
• For \(2x - 5 > x - 7\), we did the same, leading to \(x > -2\).

After finding solutions for each, we combined them. Since \(x > -2\) is more restrictive (as -2 is greater than -6), we deduced that \(x > -2\) is the true solution. This process demonstrates how careful algebraic manipulation is key to finding the correct range of values that satisfy a compound inequality.