Problem 29 - College Algebra

Question

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 \(x > -2\) or in interval notation, \((-2, \infty)\).

Step 1: Break Down the Compound Inequality

The compound inequality given is \( 3x + 1 > 2x - 5 > x - 7 \). This can be seen as two separate inequalities to solve: \( 3x + 1 > 2x - 5 \) and \( 2x - 5 > x - 7 \). We'll solve each one separately.

Step 2: Solve the First Inequality

For the inequality \( 3x + 1 > 2x - 5 \), subtract \( 2x \) from both sides to isolate \( x \) on one side: \( 3x - 2x + 1 > -5 \) gives us \( x + 1 > -5 \). Now, 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 to isolate terms with \( x \): \( 2x - x - 5 > -7 \) gives us \( x - 5 > -7 \). Add \( 5 \) to both sides: \( x > -2 \).

Step 4: Combine the Solutions

Combine the solutions from the two inequalities: \( x > -6 \) and \( x > -2 \). The more restrictive condition is \( x > -2 \), since \( -2 \) is greater than \( -6 \). Therefore, the combined solution in inequality form is \( x > -2 \).

Step 5: Express the Solution in Interval Notation

Express the solution \( x > -2 \) in interval notation. The interval notation for this solution is \((-2, \infty)\).

Key Concepts

Inequalities

Inequalities are mathematical expressions used to show that one quantity is larger or smaller than another. Instead of using an equal sign like in equations, inequalities use special symbols.

• "Greater than" symbol (>): indicates that one number is larger than another.
• "Less than" symbol (<): used when a number is smaller.
• "Greater than or equal to" (≥) and "less than or equal to" (≤) indicate that the numbers can also be equal.

When you're dealing with inequalities, it's often important to remember that the direction of the inequality sign flips if you multiply or divide by a negative number. This is an essential rule that can change the solution completely. Compound inequalities involve more than one inequality joined together by words like "and" or "or." For example, the problem above, encompassing two connected inequalities, uses such a compound format. Solving these inequalities helps us find a range of values that satisfy all parts of the inequality.

Interval Notation

Interval notation is a way of expressing a set of numbers where all numbers between two endpoints are included.
This method transforms solutions of inequalities into a concise format. In interval notation, we use brackets and parentheses to show inclusivity or exclusivity of the endpoints:

• "(" and ")" for values that are not included, indicating an open interval.
• "[" and "]" for values that are included, indicating a closed interval.

In the solution from the exercise, we ended up with an inequality of the form \( x > -2 \).
This inequality is expressed in interval notation as \((-2, \infty)\), an open interval where \(-2\) is not included and it stretches towards infinity on the positive side, implying all values greater than \(-2\).

Solving Inequalities

The process of solving inequalities is quite similar to solving equations, with a few important differences. Like equations, you manipulate the inequality to isolate the variable, performing mathematical operations such as addition, subtraction, multiplication, or division. However, one significant distinction when solving inequalities is the "flipping" rule. If you ever multiply or divide the inequality by a negative number, you'll need to reverse the inequality sign. This ensures the inequality remains balanced.

In the exercise we tackled

• Step 1: Began by breaking the compound inequality into two parts.
• Step 2: Solved each inequality independently until the variable was isolated.
• Step 4: Combined the results to find the common solution.

Always double-check your solution by substituting possible values back into the original inequality. This confirms the correctness of your solution set.

Algebraic Expressions

Algebraic expressions contain numbers, variables, and operations like addition, subtraction, multiplication, and division.
These expressions form the basis for creating equations and inequalities.In solving the inequality problem, our task involved manipulating algebraic expressions accurately. For instance, when we had the expression \( 3x + 1 > 2x - 5 \), we performed operations to isolate the variable \( x \) on one side of the inequality:

• First, subtract terms with \( x \) from both sides if needed.
• Then, simplify the remaining terms.
• Lastly, solve for the variable.

Understanding how to manipulate these expressions is crucial when working with inequalities or any form of algebra. Remember, correctly simplifying these expressions ensures the solution you derive is valid and accurate.