To solve the compound inequality [tex]\(-\frac{1}{2} \leq \frac{1}{4} x + 1 < \frac{1}{2}\)[/tex], we need to break it down into two separate inequalities and solve each one step-by-step. Here are those steps:
### First Inequality:
[tex]\[
-\frac{1}{2} \leq \frac{1}{4}x + 1
\][/tex]
1. Subtract 1 from both sides of the inequality:
[tex]\[
-\frac{1}{2} - 1 \leq \frac{1}{4}x
\][/tex]
2. Simplify:
[tex]\[
-\frac{3}{2} \leq \frac{1}{4}x
\][/tex]
3. Multiply both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
-6 \leq x
\][/tex]
### Second Inequality:
[tex]\[
\frac{1}{4}x + 1 < \frac{1}{2}
\][/tex]
1. Subtract 1 from both sides of the inequality:
[tex]\[
\frac{1}{4}x + 1 - 1 < \frac{1}{2} - 1
\][/tex]
2. Simplify:
[tex]\[
\frac{1}{4}x < -\frac{1}{2}
\][/tex]
3. Multiply both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[
x < -2
\][/tex]
### Combine the Solutions:
Now we have two inequalities:
[tex]\[
-6 \leq x \quad \text{and} \quad x < -2
\][/tex]
This means [tex]\(x\)[/tex] is within the range:
[tex]\[
-6 \leq x < -2
\][/tex]
### Graphing the Solution:
- This is a closed interval at [tex]\(x = -6\)[/tex] (inclusive of [tex]\(-6\)[/tex]) indicated by a solid dot.
- This is an open interval at [tex]\(x = -2\)[/tex] (exclusive of [tex]\(-2\)[/tex]) indicated by an open circle.
- The solution set on the number line is the interval from [tex]\(-6\)[/tex] to [tex]\(-2\)[/tex].
To graph this on a number line:
1. Place a solid dot at [tex]\(x = -6\)[/tex].
2. Place an open circle at [tex]\(x = -2\)[/tex].
3. Draw a line connecting the two points to indicate all values between [tex]\(-6\)[/tex] and [tex]\(-2\)[/tex].
Visually, it looks like this:
[tex]\[
\bullet \quad \leftrightarrow \quad \circ
\][/tex]
So the interval [tex]\([-6, -2)\)[/tex] represents the solution set for the inequality.
You should mark this interval appropriately on the given number line by using the appropriate tool to show the range of numbers that satisfy the inequality.
