Problem 29 - Precalculus Student Solutions Manual 5th

Question

Solve each inequality. Write each solution set in interval notation. $$-5<5+2 x<11$$

Step-by-Step Solution

Answer

The solution set in interval notation is \((-5, 3)\).

Step 1: Break Down the Double Inequality

Start by breaking the double inequality \(-5 < 5 + 2x < 11\) into two separate inequalities.1. \(-5 < 5 + 2x\)2. \(5 + 2x < 11\)

Step 2: Solve the First Inequality

Solve the first inequality \(-5 < 5 + 2x\). Subtract 5 from both sides: \(-5 - 5 < 2x \)This simplifies to \(-10 < 2x\).Next, divide both sides by 2: \(\frac{-10}{2} < x \) This simplifies to \(-5 < x\)

Step 3: Solve the Second Inequality

Solve the second inequality \(5 + 2x < 11\). Subtract 5 from both sides: \(2x < 11 - 5\)This simplifies to \(2x < 6\). Next, divide both sides by 2: \(x < \frac{6}{2}\) This simplifies to \(x < 3\)

Step 4: Combine the Solutions

Combine the solutions from the two separate inequalities: \(-5 < x < 3\).

Step 5: Write in Interval Notation

Express the solution in interval notation: \((-5, 3)\).

Key Concepts

Double Inequality

A double inequality is a shorthand way to represent two separate inequalities that share a common variable.
In this exercise, the double inequality is \(-5 < 5 + 2x < 11\).
This represents two inequalities:

• \(-5 < 5 + 2x\)
• \(5 + 2x < 11\)

Breaking down a double inequality into individual inequalities makes it easier to solve.
By solving each part separately, you can then combine the solutions to get the answer to the original double inequality.
This step-by-step method is essential for correctly tackling such problems.

Solving Linear Inequalities

Solving linear inequalities involves similar steps to solving linear equations, but with extra attention to the direction of the inequality sign.
Here's how to do it:

• Isolate the variable on one side of the inequality.
• If you divide or multiply by a negative number, remember to flip the inequality sign.

For example, to solve \(-5 < 5 + 2x\):

• Subtract 5 from both sides: \(-10 < 2x\).
• Divide both sides by 2: \-5 < x\.

Notice that we didn't need to flip the sign because we didn't divide by a negative number.
Repeat this method for the second inequality, and you will get the combined solution.

Interval Notation

Interval notation is a way to describe the set of solutions to an inequality.
Instead of writing \-5 < x < 3\, we use interval notation to make it more concise:

• We use parentheses \(()\) to denote that the endpoints are not included.

So \-5 < x < 3\ is written as \((-5, 3)\).
Understanding interval notation helps clearly communicate the range of possible values for a variable.
Keep in mind that:

• \(()\) means the endpoints are not included.
• \([]\) means the endpoints are included.
• You can combine both, like \((-\infty, 2]\), for mixed intervals.

Using interval notation is an efficient and standardized way to express solutions.