Problem 25 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ x-3 \leq 6 \text { and } x+2 \geq 7 $$

Step-by-Step Solution

Answer

[5, 9]

Step 1 - Solve the first inequality

Start with the first inequality: \(x - 3 \leq 6\)Add 3 to both sides to isolate \(x\): \[x - 3 + 3 \leq 6 + 3\]Simplifying this, we get: \[x \leq 9\]

Step 2 - Solve the second inequality

Now solve the second inequality: \(x + 2 \geq 7\)Subtract 2 from both sides to isolate \(x\): \[x + 2 - 2 \geq 7 - 2\]Simplifying this, we get: \[x \geq 5\]

Step 3 - Combine the inequalities

Now combine the two results: \(x \leq 9\) and \(x \geq 5\)This can be written as a compound inequality: \[5 \leq x \leq 9\]

Step 4 - Write the solution in interval notation

The solution in interval notation is: \[\[5, 9\]\]

Step 5 - Graph the solution set

To graph the solution set on a number line, draw a number line with points at 5 and 9. Shade the region between 5 and 9, including the endpoints because it is a closed interval. The graph looks like this: \[\text{---[====5====9]---}\]

Key Concepts

solving inequalities

Solving inequalities involves finding all the values of a variable that make the inequality true. To isolate the variable, we perform operations similar to those we use for solving equations. These operations should keep the inequality balanced. For example, let's solve the first part of our compound inequality:

Step-by-step, we work with:\( x - 3 \leq 6 \). To isolate \(x\), add 3 to both sides:
\(x - 3 + 3 \leq 6 + 3 \). This simplifies to \(x \leq 9\).

Next, we solve the second inequality:
\( x + 2 \geq 7 \)
Subtract 2 from both sides to isolate \(x\): \(x + 2 - 2 \geq 7 - 2\)
Simplifying this, we get: \(x \geq 5\).

Combining these two results, we get: \(5 \leq x \leq 9\).

interval notation

Interval notation is a way to describe the set of solutions for inequalities. It shows the range of values that solve the inequality using brackets and parentheses. Here is how it works:

• Brackets [ ] denote that the endpoint values are included in the interval. This is known as a closed interval.
• Parentheses ( ) denote that the endpoint values are not included. This is known as an open interval.

For the inequality \(5 \leq x \leq 9\), both endpoints are included because the inequality is `less than or equal to` and `greater than or equal to`.
Therefore, in interval notation, we write this as: \[ [5, 9] \]

graphing solutions

Graphing solutions on a number line helps to visually represent the set of all possible solutions to an inequality. The key steps to graphing an inequality include:

• Drawing a number line.
• Identifying and marking the endpoints.
• Shading the region that represents the solution set.

For our compound inequality \(5 \leq x \leq 9\), we mark the points 5 and 9 on the number line. Since these endpoints are included in the solution (closed interval), we use solid dots.
Then we shade the region between 5 and 9 to show that all values in this interval satisfy the compound inequality. The final graph looks like this:

\[ \text{---[===5===9===]---} \]