To determine the values of [tex]\( x \)[/tex] that satisfy both inequalities [tex]\( 9 + 2x > 0 \)[/tex] and [tex]\( 7 - 3x > 0 \)[/tex], we will solve each inequality step-by-step and then find the common values that satisfy both.
### Solving the First Inequality:
[tex]\[ 9 + 2x > 0 \][/tex]
1. Isolate [tex]\( x \)[/tex]:
[tex]\[ 2x > -9 \][/tex]
2. Divide both sides by 2:
[tex]\[ x > -\frac{9}{2} \][/tex]
So, the solution for the first inequality is:
[tex]\[ x > -4.5 \][/tex]
### Solving the Second Inequality:
[tex]\[ 7 - 3x > 0 \][/tex]
1. Isolate [tex]\( x \)[/tex]:
[tex]\[ -3x > -7 \][/tex]
2. Divide both sides by -3 and reverse the inequality sign (since we are dividing by a negative number):
[tex]\[ x < \frac{7}{3} \][/tex]
So, the solution for the second inequality is:
[tex]\[ x < \frac{7}{3} \][/tex]
### Finding the Intersection:
To find the common values that satisfy both inequalities, we need to determine the values of [tex]\( x \)[/tex] that lie within both ranges:
[tex]\[ x > -4.5 \][/tex]
[tex]\[ x < \frac{7}{3} \][/tex]
The intersection is the set of values of [tex]\( x \)[/tex] that satisfy both conditions simultaneously. These values are in the range:
[tex]\[ -4.5 < x < \frac{7}{3} \][/tex]
### Conclusion:
The values of [tex]\( x \)[/tex] that satisfy both inequalities [tex]\( 9 + 2x > 0 \)[/tex] and [tex]\( 7 - 3x > 0 \)[/tex] are:
[tex]\[ -4.5 < x < \frac{7}{3} \][/tex]
This is the solution set for the given inequalities.
