Problem 50 - Algebra for College Students

Question

Solve each compound inequality using the compact form. Express the solution sets in interval notation. \(0<2 x+5<12\)

Step-by-Step Solution

Answer

The solution in interval notation is \((-\frac{5}{2}, \frac{7}{2})\).

Step 1: Break down the compound inequality

The compound inequality is given as \(0 < 2x + 5 < 12\). This is actually two separate inequalities that must be true simultaneously: \(0 < 2x + 5\) and \(2x + 5 < 12\).

Step 2: Solve the first inequality

To solve \(0 < 2x + 5\), subtract 5 from both sides to isolate the term with \(x\):\[ 0 - 5 < 2x + 5 - 5 \]This simplifies to:\[ -5 < 2x \]Now, divide both sides by 2 to solve for \(x\):\[ \frac{-5}{2} < x \]

Step 3: Solve the second inequality

For the inequality \(2x + 5 < 12\), subtract 5 from both sides:\[ 2x + 5 - 5 < 12 - 5 \]This simplifies to:\[ 2x < 7 \]Now, divide both sides by 2 to solve for \(x\):\[ x < \frac{7}{2} \]

Step 4: Write the combined solution

The solution set for the compound inequality requires that both conditions \(\frac{-5}{2} < x\) and \(x < \frac{7}{2}\) must be satisfied at the same time. This can be expressed as:\[ \frac{-5}{2} < x < \frac{7}{2} \]

Step 5: Express the solution in interval notation

To express the solution \(\frac{-5}{2} < x < \frac{7}{2}\) in interval notation, it is written as the interval going from \(-\frac{5}{2}\) to \(\frac{7}{2}\):(\(-\frac{5}{2}, \frac{7}{2}\))

Key Concepts

Interval Notation

When solving compound inequalities, one of the main goals is to express the solution in a manner that is easy to read and understand. This is where interval notation becomes particularly useful. Interval notation provides a way to represent a set of numbers where all elements satisfy the inequality. It is like giving a starting point and an ending point for all numbers that are part of the solution.

In our exercise, we ended up with the solution \(\frac{-5}{2} < x < \frac{7}{2}\). That's a range of values for \(x\). Interval notation expresses this range as \((-\frac{5}{2}, \frac{7}{2})\).

This notation uses:

• Round parentheses ( ) to show that the endpoints are not included in the set.
• If an endpoint should be included, a square bracket [ ] is used instead.

This simple notation makes it easy to see the values of \(x\) that are allowed by the inequality.

Solving Inequalities

Solving inequalities is about finding the range of possible values for a variable that makes the inequality true. Unlike equations where we find a specific value, inequalities encompass a range of values.

To solve a compound inequality like \(0 < 2x + 5 < 12\), break it down into two separate inequalities:

• \(0 < 2x + 5\) and
• \(2x + 5 < 12\)

By addressing these individually, you can find multiple conditions that \(x\) must satisfy.

Remember these important steps:

• Isolate the variable on one side using basic operations.
• If you multiply or divide both sides by a negative number, reverse the inequality sign.

In our exercise, no negative multiplication or division was required, so the inequality signs remained the same throughout the process.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying inequalities or equations to isolate the variable. It is an essential skill for solving inequalities efficiently.

When solving the compound inequality \(0 < 2x + 5 < 12\):

• Start by subtracting 5 from all parts of the inequality to eliminate the constant on the side with the variable.
• This step simplifies to \(-5 < 2x < 7\).
• Then, divide by 2 to solve for \(x\), giving \(-\frac{5}{2} < x < \frac{7}{2}\).

It's crucial to perform the same operations on every part of a compound inequality to maintain its balance.

Understanding how to manipulate algebraic expressions is key to solving not just inequalities but a wide variety of mathematical problems. It requires practice but can greatly simplify what might initially seem like complex mathematical tasks.