To find the lowest integer value that [tex]\( r \)[/tex] can take for the given inequality [tex]\( 7r + 5 > 37 - 3r \)[/tex], follow these steps:
1. Isolate the variable [tex]\( r \)[/tex]:
Start with the given inequality:
[tex]\[
7r + 5 > 37 - 3r
\][/tex]
To isolate [tex]\( r \)[/tex], move all terms involving [tex]\( r \)[/tex] to one side. Add [tex]\( 3r \)[/tex] to both sides:
[tex]\[
7r + 3r + 5 > 37
\][/tex]
Combine the [tex]\( r \)[/tex] terms on the left side:
[tex]\[
10r + 5 > 37
\][/tex]
2. Move the constant term:
Subtract 5 from both sides to isolate terms involving [tex]\( r \)[/tex]:
[tex]\[
10r > 37 - 5
\][/tex]
Simplify the right side:
[tex]\[
10r > 32
\][/tex]
3. Solve for [tex]\( r \)[/tex]:
Divide both sides by 10 to solve for [tex]\( r \)[/tex]:
[tex]\[
r > \frac{32}{10}
\][/tex]
Simplify the division:
[tex]\[
r > 3.2
\][/tex]
4. Find the lowest integer value greater than 3.2:
The lowest integer value that is greater than 3.2 is 4.
Therefore, the lowest integer value that [tex]\( r \)[/tex] can take is:
[tex]\[
r = 4
\][/tex]
So, the detailed step-by-step solution shows that the smallest integer [tex]\( r \)[/tex] can be for the inequality [tex]\( 7r + 5 > 37 - 3r \)[/tex] is 4.
