Question

Solve the following inequality for rr. Write your answer in simplest form.
5, r, minus, 5, left parenthesis, minus, 7, r, plus, 9, right parenthesis, is greater than or equal to, minus, 10, r, plus, 9, minus, 6, r
5r−5(−7r+9)≥
−10r+9−6r

Answer 1

To solve the inequality \( 5r - 5(-7r + 9) \geq -10r + 9 - 6r \), let's simplify each side step by step.

Starting with the left side:

\[
5r - 5(-7r + 9)
\]
Distributing \( -5 \):

\[
5r + 35r - 45
\]
Combine like terms:

\[
(5r + 35r) - 45 = 40r - 45
\]

Now the left side has become \( 40r - 45 \).

Next, simplify the right side:

\[
-10r + 9 - 6r
\]
Combine like terms:

\[
(-10r - 6r) + 9 = -16r + 9
\]

Now we can write the entire inequality as:

\[
40r - 45 \geq -16r + 9
\]

Next, let's move all terms involving \( r \) to one side and constant terms to the other side. We can do this by adding \( 16r \) to both sides and adding \( 45 \) to both sides:

\[
40r + 16r \geq 9 + 45
\]
Combine like terms:

\[
56r \geq 54
\]

Now, divide both sides by \( 56 \):

\[
r \geq \frac{54}{56}
\]

Simplifying \( \frac{54}{56} \):

\[
\frac{54 \div 2}{56 \div 2} = \frac{27}{28}
\]

Thus, the solution to the inequality is:

\[
r \geq \frac{27}{28}
\]