To solve the inequality [tex]\(2.9(x + 8) < 26.1\)[/tex] and determine the solution set, follow these detailed steps:
### Step 1: Distribute the 2.9
First, we need to distribute the 2.9 inside the parenthesis:
[tex]\[ 2.9(x + 8) = 2.9 \cdot x + 2.9 \cdot 8 \][/tex]
[tex]\[ 2.9x + 23.2 \][/tex]
So the inequality becomes:
[tex]\[ 2.9x + 23.2 < 26.1 \][/tex]
### Step 2: Isolate the variable term
Next, we isolate the term with the variable [tex]\(x\)[/tex]. To do this, subtract 23.2 from both sides of the inequality:
[tex]\[ 2.9x + 23.2 - 23.2 < 26.1 - 23.2 \][/tex]
[tex]\[ 2.9x < 2.9 \][/tex]
### Step 3: Solve for [tex]\(x\)[/tex]
Now, we need to solve for [tex]\(x\)[/tex] by dividing both sides of the inequality by 2.9:
[tex]\[ \frac{2.9x}{2.9} < \frac{2.9}{2.9} \][/tex]
[tex]\[ x < 1 \][/tex]
### Conclusion:
The solution set of the inequality is [tex]\( x < 1 \)[/tex].
### Graphing the Solution Set:
1. Draw a number line.
2. Mark the point [tex]\( x = 1 \)[/tex] on the number line.
3. Since the inequality is strictly less than ([tex]\( < \)[/tex]) and not less than or equal to ([tex]\( \leq \)[/tex]), we use an open circle at [tex]\( x = 1 \)[/tex] to indicate that [tex]\( x = 1 \)[/tex] is not included in the solution set.
4. Shade the region to the left of [tex]\( x = 1 \)[/tex] to represent all the values of [tex]\( x \)[/tex] that are less than 1.
The graph of the solution set will show an open circle at [tex]\( x = 1 \)[/tex] with shading extending to the left along the number line.
