To find the integers that satisfy both inequalities \(2x + 9 < 0\) and \(x > -12\), follow these detailed steps:
### 1. Solve the first inequality: \(2x + 9 < 0\)
First, isolate \(x\):
[tex]\[
2x + 9 < 0
\][/tex]
Subtract 9 from both sides:
[tex]\[
2x < -9
\][/tex]
Divide both sides by 2:
[tex]\[
x < -\frac{9}{2}
\][/tex]
Since \(-\frac{9}{2}\) is equivalent to \(-4.5\), we have:
[tex]\[
x < -4.5
\][/tex]
### 2. Solve the second inequality: \(x > -12\)
This inequality is already solved:
[tex]\[
x > -12
\][/tex]
### 3. Combine the inequalities
Now we need to find the intersection of these two solutions:
[tex]\[
-12 < x < -4.5
\][/tex]
### 4. Identify the integers in the solution set
Integers are whole numbers. Within the interval \(-12 < x < -4.5\), the integers are:
[tex]\[
-11, -10, -9, -8, -7, -6, -5
\][/tex]
Thus, the integers that satisfy both inequalities are:
[tex]\[
-11, -10, -9, -8, -7, -6, -5
\][/tex]
