Answer :
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]
### 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]