To solve the inequality [tex]\(2.9(x + 8) < 26.1\)[/tex], we will go through the steps systematically.
1. Distribute the 2.9:
[tex]\[
2.9(x + 8) = 2.9x + 2.9 \cdot 8
\][/tex]
Simplifying this, we get:
[tex]\[
2.9x + 23.2
\][/tex]
2. Write the inequality:
[tex]\[
2.9x + 23.2 < 26.1
\][/tex]
3. Isolate the term with the variable [tex]\(x\)[/tex]:
Subtract 23.2 from both sides of the inequality:
[tex]\[
2.9x + 23.2 - 23.2 < 26.1 - 23.2
\][/tex]
Simplifying this, we get:
[tex]\[
2.9x < 2.9
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides of the inequality by 2.9:
[tex]\[
\frac{2.9x}{2.9} < \frac{2.9}{2.9}
\][/tex]
Simplifying this, we get:
[tex]\[
x < 1
\][/tex]
Thus, the solution to the inequality [tex]\(2.9(x + 8) < 26.1\)[/tex] is [tex]\(x < 1\)[/tex].
Interpreting Graphically:
The solution set [tex]\(x < 1\)[/tex] means we are looking for all values of [tex]\(x\)[/tex] that are less than 1.
On a number line or graph:
- Draw a number line.
- Locate the point [tex]\(1\)[/tex] on the number line.
- Draw an open circle at [tex]\(1\)[/tex] to indicate that [tex]\(1\)[/tex] is not included in the solution.
- Shade the region to the left of [tex]\(1\)[/tex] to represent all values less than [tex]\(1\)[/tex].
So, the graph showing the solution set will have a number line with an open circle at [tex]\(x = 1\)[/tex] and shading to the left of this point.
