Question

Solve the inequality. Then graph the solution and give interval notation.
2≤-2x+6<12
-5,
-4,
-3,
-2,
-1
0
1
2
3
4
5
Interval notation for the above graph and inequality is

Answer 1

To solve the compound inequality \( 2 \leq -2x + 6 < 12 \), we'll break it down into two parts and solve each part separately.

### Step 1: Solve the first part \( 2 \leq -2x + 6 \)

1. Subtract 6 from both sides:
\[
2 - 6 \leq -2x
\]
\[
-4 \leq -2x
\]

2. Divide both sides by -2 (remember to flip the inequality sign):
\[
\frac{-4}{-2} \geq x
\]
\[
2 \geq x \quad \text{or} \quad x \leq 2
\]

### Step 2: Solve the second part \( -2x + 6 < 12 \)

1. Subtract 6 from both sides:
\[
-2x < 12 - 6
\]
\[
-2x < 6
\]

2. Divide both sides by -2 (again, flip the inequality sign):
\[
x > \frac{6}{-2}
\]
\[
x > -3
\]

### Step 3: Combine the results

From the two parts, we have:
- \( x \leq 2 \)
- \( x > -3 \)

Combining these gives us:
\[
-3 < x \leq 2
\]

### Step 4: Interval Notation

The solution in interval notation is:
\[
(-3, 2]
\]

### Step 5: Graphing

The graph will include an open circle at -3 (not included) and a solid circle at 2 (included), with a line connecting the two.

### Final answer:

The interval notation for the solution is:
\[
(-3, 2]
\]