To determine which are correct representations of the inequality [tex]\(6x \geq 3 + 4(2x - 1)\)[/tex], let's solve the inequality step-by-step.
1. Start with the given inequality:
[tex]\[
6x \geq 3 + 4(2x - 1)
\][/tex]
2. Distribute the 4 on the right side:
[tex]\[
4(2x - 1) = 4 \cdot 2x - 4 \cdot 1 = 8x - 4
\][/tex]
3. Substitute this back into the inequality:
[tex]\[
6x \geq 3 + 8x - 4
\][/tex]
4. Combine like terms on the right side:
[tex]\[
6x \geq 8x - 1
\][/tex]
5. To isolate [tex]\(x\)[/tex], subtract [tex]\(8x\)[/tex] from both sides:
[tex]\[
6x - 8x \geq -1
\][/tex]
[tex]\[
-2x \geq -1
\][/tex]
6. To solve for [tex]\(x\)[/tex], divide both sides by [tex]\(-2\)[/tex] and remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
x \leq \frac{1}{2}
\][/tex]
So, the solution to the inequality is [tex]\(x \leq \frac{1}{2}\)[/tex].
Now, let's verify which representations are correct:
1. [tex]\(1 \geq 2x\)[/tex]
This representation simplifies to:
[tex]\[
\frac{1}{2} \geq x \quad \text{or} \quad x \leq \frac{1}{2}
\][/tex]
This is indeed correct and matches our solution.
2. [tex]\(6x \geq 3 + 8x - 4\)[/tex]
Simplifying inside:
[tex]\[
3 + 8x - 4 = 8x - 1
\][/tex]
Which gives us:
[tex]\[
6x \geq 8x - 1
\][/tex]
This is also correct and matches our earlier steps.
Thus, the correct representations of the inequality are:
- [tex]\(1 \geq 2x\)[/tex]
- [tex]\(6x \geq 3 + 8x - 4\)[/tex]
We have successfully found the two correct representations of the inequality.
