Question

Which linear equation has the same solution set as 9d + 3 - 2d = -2(3 + d)?(1 point) Responses 10d + 2 = 27 10d + 2 = 27 5d = 10 5d = 10 9d = -9 9d = -9 2d = 2d
Which option is the solution to the inequality -12g + 18 > -5g + 53?(1 point) Responses g > -5 g > -5 g < 71 g < 71 g < -5 g < -5 g > 71

Answer 1

Let's solve each question one by one.

**1. Solve the equation \(9d + 3 - 2d = -2(3 + d)\):**

First, simplify the left side:
\[
9d - 2d + 3 = 7d + 3
\]

Now simplify the right side:
\[
-2(3 + d) = -6 - 2d
\]

Now, set the two simplified expressions equal to each other:
\[
7d + 3 = -6 - 2d
\]

Next, add \(2d\) to both sides:
\[
7d + 2d + 3 = -6
\]
\[
9d + 3 = -6
\]

Now subtract \(3\) from both sides:
\[
9d = -9
\]

Finally, divide by \(9\):
\[
d = -1
\]

To find the equivalent equation, we can rewrite \(9d = -9\) as it shows \(d\)'s equivalent relationship.

Now let's check the provided options:

- \(10d + 2 = 27\) — Not equivalent.
- \(5d = 10\) — Not equivalent.
- \(9d = -9\) — This is equivalent.
- \(2d = 2d\) — This is always true, but not informative.

Thus, the correct equivalent equation is:
**9d = -9.**

---

**2. Solve the inequality \(-12g + 18 > -5g + 53\):**

First, simplify by isolating \(g\):
Add \(12g\) to both sides:
\[
18 > 7g + 53
\]

Now subtract \(53\) from both sides:
\[
18 - 53 > 7g
\]
\[
-35 > 7g
\]

Now, divide both sides by \(7\):
\[
-5 > g
\]

Rearranging gives:
\[
g < -5
\]

Now let's check the provided options:

- \(g > -5\) — Incorrect.
- \(g < 71\) — Incorrect, does not correspond to our solution.
- \(g < -5\) — Correct.
- \(g > 71\) — Incorrect.

Thus, the correct solution to the inequality is:
**g < -5.**