Let's solve the inequality step-by-step to find the solution set.
The problem states:
Five times the sum of a number and 27 is greater than or equal to six times the sum of that number and 26. We need to find the range of values for this number.
Let's denote the unknown number as [tex]\( x \)[/tex].
1. Set up the inequality:
Five times the sum of a number and 27:
[tex]\[
5(x + 27)
\][/tex]
Six times the sum of a number and 26:
[tex]\[
6(x + 26)
\][/tex]
The inequality can be expressed as:
[tex]\[
5(x + 27) \geq 6(x + 26)
\][/tex]
2. Expand both sides of the inequality:
[tex]\[
5x + 135 \geq 6x + 156
\][/tex]
3. Isolate [tex]\( x \)[/tex] on one side of the inequality:
Subtract [tex]\( 5x \)[/tex] from both sides:
[tex]\[
135 \geq x + 156
\][/tex]
Subtract 156 from both sides:
[tex]\[
135 - 156 \geq x
\][/tex]
Simplify:
[tex]\[
-21 \geq x
\][/tex]
This can be rewritten as:
[tex]\[
x \leq -21
\][/tex]
Therefore, the solution set for [tex]\( x \)[/tex] is:
[tex]\[
(-\infty, -21]
\][/tex]
So, the correct answer is:
[tex]\[
(-\infty, -21]
\][/tex]
