Answer :
To determine which graph shows the solution set of the inequality [tex]\( 2.9(x + 8) < 26.1 \)[/tex], we can solve it step-by-step.
1. Distribute the 2.9 on the left-hand side:
[tex]\[ 2.9(x + 8) \Rightarrow 2.9 \cdot x + 2.9 \cdot 8 \Rightarrow 2.9x + 23.2 \][/tex]
So the inequality becomes:
[tex]\[ 2.9x + 23.2 < 26.1 \][/tex]
2. Isolate the [tex]\( x \)[/tex] term by subtracting 23.2 from both sides:
[tex]\[ 2.9x + 23.2 - 23.2 < 26.1 - 23.2 \][/tex]
Simplifying this gives:
[tex]\[ 2.9x < 2.9 \][/tex]
3. Solve for [tex]\( x \)[/tex] by dividing both sides by 2.9:
[tex]\[ \frac{2.9x}{2.9} < \frac{2.9}{2.9} \][/tex]
This simplifies to:
[tex]\[ x < 1 \][/tex]
The solution to the inequality is [tex]\( x < 1 \)[/tex].
To graph this solution on a number line:
- Draw a horizontal number line.
- Locate the number 1 on the number line.
- Shade the region to the left of 1 to represent all numbers less than 1.
- Place an open circle or hollow dot at 1 to show that 1 is not included in the solution.
In summary, the graph that correctly represents the solution set of the inequality [tex]\( 2.9(x + 8) < 26.1 \)[/tex] will have an open circle at [tex]\( x = 1 \)[/tex] and shading to the left, indicating all values of [tex]\( x \)[/tex] that are less than 1.
1. Distribute the 2.9 on the left-hand side:
[tex]\[ 2.9(x + 8) \Rightarrow 2.9 \cdot x + 2.9 \cdot 8 \Rightarrow 2.9x + 23.2 \][/tex]
So the inequality becomes:
[tex]\[ 2.9x + 23.2 < 26.1 \][/tex]
2. Isolate the [tex]\( x \)[/tex] term by subtracting 23.2 from both sides:
[tex]\[ 2.9x + 23.2 - 23.2 < 26.1 - 23.2 \][/tex]
Simplifying this gives:
[tex]\[ 2.9x < 2.9 \][/tex]
3. Solve for [tex]\( x \)[/tex] by dividing both sides by 2.9:
[tex]\[ \frac{2.9x}{2.9} < \frac{2.9}{2.9} \][/tex]
This simplifies to:
[tex]\[ x < 1 \][/tex]
The solution to the inequality is [tex]\( x < 1 \)[/tex].
To graph this solution on a number line:
- Draw a horizontal number line.
- Locate the number 1 on the number line.
- Shade the region to the left of 1 to represent all numbers less than 1.
- Place an open circle or hollow dot at 1 to show that 1 is not included in the solution.
In summary, the graph that correctly represents the solution set of the inequality [tex]\( 2.9(x + 8) < 26.1 \)[/tex] will have an open circle at [tex]\( x = 1 \)[/tex] and shading to the left, indicating all values of [tex]\( x \)[/tex] that are less than 1.
