Certainly! Let's solve each inequality step-by-step and then combine their solutions on a number line.
### Inequality 1: [tex]\( x + 4 > 9 \)[/tex]
1. Isolate [tex]\( x \)[/tex] by subtracting 4 from both sides of the inequality:
[tex]\[
x + 4 - 4 > 9 - 4
\][/tex]
2. Simplify:
[tex]\[
x > 5
\][/tex]
### Inequality 2: [tex]\( 2x - 3 < 7 \)[/tex]
1. Isolate [tex]\( x \)[/tex] by first adding 3 to both sides of the inequality:
[tex]\[
2x - 3 + 3 < 7 + 3
\][/tex]
2. Simplify:
[tex]\[
2x < 10
\][/tex]
3. Divide both sides by 2:
[tex]\[
x < 5
\][/tex]
### Combine the Solutions
The solution to the first inequality gives us:
[tex]\[ x > 5 \][/tex]
The solution to the second inequality gives us:
[tex]\[ x < 5 \][/tex]
### Final Solution Set
To find the intersection (overlap) of the two inequalities, we look at where [tex]\( x \)[/tex] must satisfy both conditions simultaneously. From the solutions above, we see that:
[tex]\[ x > 5 \][/tex] and [tex]\[ x < 5 \][/tex] do not overlap.
Thus, there is no value of [tex]\( x \)[/tex] that satisfies both inequalities simultaneously. Therefore, the combined solution set is empty.
### Graph the Solution Set
To graph the solution set for each inequality separately:
1. For [tex]\( x > 5 \)[/tex]:
- Draw an open circle at [tex]\( x = 5 \)[/tex] and shade all numbers to the right of 5 (since it includes numbers greater than 5 but not 5 itself).
2. For [tex]\( x < 5 \)[/tex]:
- Draw an open circle at [tex]\( x = 5 \)[/tex] and shade all numbers to the left of 5 (since it includes numbers less than 5 but not 5 itself).
When you place both graphs on the same number line, you'll notice there is no overlapping region where [tex]\( x \)[/tex] can be both greater than 5 and less than 5 at the same time.
### Conclusion
There is no number [tex]\( x \)[/tex] that can satisfy both [tex]\( x + 4 > 9 \)[/tex] and [tex]\( 2x - 3 < 7 \)[/tex] simultaneously. Hence, the solution set is empty.
