To determine which graph represents the solution set of the inequality [tex]\(2.9(x+8) < 26.1\)[/tex], let's follow these steps:
1. Identify the inequality and start solving:
The given inequality is:
[tex]\[
2.9(x + 8) < 26.1
\][/tex]
2. Distribute the 2.9 inside the parentheses:
[tex]\[
2.9 \cdot x + 2.9 \cdot 8 < 26.1
\][/tex]
Simplifying the arithmetic inside the distribution:
[tex]\[
2.9x + 23.2 < 26.1
\][/tex]
3. Isolate the variable [tex]\(x\)[/tex]:
Subtract 23.2 from both sides of the inequality:
[tex]\[
2.9x < 26.1 - 23.2
\][/tex]
Perform the subtraction on the right-hand side:
[tex]\[
2.9x < 2.9
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides of the inequality by 2.9:
[tex]\[
x < \frac{2.9}{2.9}
\][/tex]
Simplify the division:
[tex]\[
x < 1
\][/tex]
The solution to the inequality [tex]\(2.9(x + 8) < 26.1\)[/tex] is:
[tex]\[
x < 1
\][/tex]
On a number line, this solution set consists of all real numbers less than 1. Since there is no lower bound explicitly defined (it extends to negative infinity), we can describe this solution set as:
[tex]\[
(-\infty, 1)
\][/tex]
On a graph, we would represent this interval as follows:
- A number line extending infinitely to the left (toward negative infinity).
- A boundary at [tex]\(x = 1\)[/tex], which is often shown as an open circle to indicate that the value 1 is not included in the solution set.
- A shaded region to the left of the open circle to indicate all values less than 1 are included in the solution set.
Therefore, the correct graph for this solution set will include:
- An open circle at [tex]\(x = 1\)[/tex].
- Shading on the number line extending to the left from [tex]\(x = 1\)[/tex].
