Problem 47 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |r+5|<20 $$

Step-by-Step Solution

Answer

(1, 11)

Step 1: Understand the Inequality

The given inequality is -4 < x - 5 < 6. This is a compound inequality and we need to solve it in parts.

Step 2: Isolate x in the Inequality

Add 5 to all parts of the inequality to isolate x: -4 + 5 < x - 5 + 5 < 6 + 5.

Step 3: Simplify the Inequality

Simplify each part: 1 < x < 11. This is the solution to the inequality.

Step 4: Graph the Solution

To graph the solution set, draw a number line and shade the region between 1 (not including 1) and 11 (not including 11). Use open circles on 1 and 11 to indicate that these values are not part of the solution.

Step 5: Write in Interval Notation

Write the solution set in interval notation: (1, 11).

Key Concepts

solving inequalities

To solve compound inequalities, you handle both parts of the inequality simultaneously. Consider the problem ewline -4 < x - 5 < 6. This inequality requires you to find an x value that makes both inequalities true.
Start by isolating the variable. In this case, adding 5 to each part of the compound inequality results in: ewline -4 + 5 < x - 5 + 5 < 6 + 5.
Simplify to achieve: 1 < x < 11. All values of x between 1 and 11 satisfy the inequality. Practicing these steps ensures you master solving more complicated inequalities.

interval notation

Once you solve an inequality, it’s useful to express the solution in interval notation. This notation provides a clear, concise way to depict the range of solutions. ewline For the given inequality 1 < x < 11, the solution set is the set of all numbers between 1 and 11, not including the boundary points. In interval notation, this is written as:
(1, 11). ewline Remember:

• ( ) denotes that the end points are not included (open interval).
• [ ] denotes that the end points are included (closed interval).

Interval notation helps to quickly communicate the solution set of the inequality in mathematical problems.

graphical representation of inequalities

Graphing inequalities visually demonstrates the solution set. Consider the inequality 1 < x < 11.
To graph this: ewline

• Draw a number line.
• Mark the points 1 and 11 with open circles to show these values are not included.
• Shade the region between 1 and 11 to indicate the solution set.

This visual representation confirms what interval notation indicates and helps cross-check your solution. ewline Practice graphing various inequalities to become proficient with this skill.

isolation of variables

Isolating the variable is a key step when solving inequalities. The objective is to have the variable on one side of the inequality. ewline For example, with the inequality -4 < x - 5 < 6, add 5 to each part to isolate x:
-4 + 5 < x - 5 + 5 < 6 + 5. Simplifying, we get ewline 1 < x < 11. ewline Isolating the variable simplifies the inequality and clarifies the relationship between the variable and the constants.
Practicing this skill strengthens your problem-solving ability in algebra and prepares you for more complex equations.