Question

Solve the inequality. Then graph the solution and give interval notation.
-10≤-4x+6<2
-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 inequality \(-10 \leq -4x + 6 < 2\), we can break it into two separate inequalities:

1. \(-10 \leq -4x + 6\)
2. \(-4x + 6 < 2\)

### Step 1: Solve the first inequality \(-10 \leq -4x + 6\)

1. Subtract 6 from both sides:
\[
-10 - 6 \leq -4x \implies -16 \leq -4x
\]

2. Divide both sides by -4. Remember to flip the inequality sign:
\[
4 \geq x \implies x \leq 4
\]

### Step 2: Solve the second inequality \(-4x + 6 < 2\)

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

2. Divide both sides by -4. Again, flip the inequality sign:
\[
x > 1
\]

### Step 3: Combine the results

From the inequalities solved above, we have:
1. \(x \leq 4\)
2. \(x > 1\)

Combining these results gives us:
\[
1 < x \leq 4
\]
In interval notation, this solution is expressed as:
\[
(1, 4]
\]

### Step 4: Graph the solution

To graph the solution \( (1, 4] \):
- Draw a number line.
- Use an open circle at \(1\) (indicating \(1\) is not included)
- Use a closed circle at \(4\) (indicating \(4\) is included).
- Shade the region between \(1\) and \(4\).

### Final Answer

The interval notation for the solution is:
\[
(1, 4]
\]