To solve the inequality [tex]\(|-4x + 6| \geq 2\)[/tex], we need to consider the properties of absolute values and analyze the two possible cases for the inequality.
The inequality [tex]\(|A| \geq B\)[/tex] implies:
1. [tex]\(A \geq B\)[/tex], or
2. [tex]\(A \leq -B\)[/tex].
Let's apply this to our problem:
[tex]\[
|-4x + 6| \geq 2
\][/tex]
This gives us two cases to consider:
### Case 1: [tex]\(-4x + 6 \geq 2\)[/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
-4x + 6 \geq 2
\][/tex]
Subtract 6 from both sides:
[tex]\[
-4x \geq 2 - 6
\][/tex]
[tex]\[
-4x \geq -4
\][/tex]
Divide by -4 (remember to reverse the inequality sign when dividing by a negative number):
[tex]\[
x \leq 1
\][/tex]
### Case 2: [tex]\(-4x + 6 \leq -2\)[/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[
-4x + 6 \leq -2
\][/tex]
Subtract 6 from both sides:
[tex]\[
-4x \leq -2 - 6
\][/tex]
[tex]\[
-4x \leq -8
\][/tex]
Divide by -4 (again, reverse the inequality sign when dividing by a negative number):
[tex]\[
x \geq 2
\][/tex]
Now we combine the solutions from Case 1 and Case 2. We have:
1. [tex]\(x \leq 1\)[/tex]
2. [tex]\(x \geq 2\)[/tex]
Thus, the solution to the inequality [tex]\(|-4x + 6| \geq 2\)[/tex] is:
[tex]\[
x \leq 1 \quad \text{or} \quad x \geq 2
\][/tex]
Looking at the provided choices:
1. [tex]\(x \geq -1\)[/tex] or [tex]\(x \leq -2\)[/tex]
2. [tex]\(1 \leq x \leq 2\)[/tex]
3. [tex]\(x \leq 1\)[/tex] or [tex]\(x \geq 2\)[/tex]
4. There are no solutions
The correct answer is:
[tex]\[
\boxed{x \leq 1 \text{ or } x \geq 2}
\][/tex]
Which corresponds to the third choice. Thus, the correct answer is:
[tex]\[
\boxed{3}
\][/tex]
