Alright, to solve the inequality where 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 can break it down step by step.
### Step 1: Set up the inequality
Let's denote the number by [tex]\( x \)[/tex]. The inequality given in the problem is:
[tex]\[ 5(x + 27) \geq 6(x + 26) \][/tex]
### Step 2: Distribute the constants
Distribute the 5 and the 6 through the parentheses:
[tex]\[ 5x + 135 \geq 6x + 156 \][/tex]
### Step 3: Move all terms involving [tex]\( x \)[/tex] to one side
To isolate [tex]\( x \)[/tex], let's move all the [tex]\( x \)[/tex]-terms to one side of the inequality. We can do this by subtracting [tex]\( 5x \)[/tex] from both sides:
[tex]\[ 135 \geq x + 156 \][/tex]
### Step 4: Move the constant terms to the other side
Next, isolate [tex]\( x \)[/tex] by moving the constants to one side. Subtract 156 from both sides:
[tex]\[ 135 - 156 \geq x \][/tex]
[tex]\[ -21 \geq x \][/tex]
or equivalently,
[tex]\[ x \leq -21 \][/tex]
### Step 5: Express the solution set
So, the solution is all [tex]\( x \)[/tex] such that:
[tex]\[ x \leq -21 \][/tex]
In interval notation, this is written as:
[tex]\[ (-\infty, -21] \][/tex]
### Conclusion
The solution set of the problem is:
[tex]\[ \boxed{(-\infty, -21]} \][/tex]
