Sure, let's solve the inequality [tex]\( 3a + 1 > \frac{-7a - 14}{-4} \)[/tex] step by step.
### Step 1: Simplify the Right-Hand Side
First, simplify the right-hand side of the inequality. Notice that the fraction has a negative denominator:
[tex]\[ \frac{-7a - 14}{-4} \][/tex]
By simplifying the expression, we can distribute the negative sign in the denominator:
[tex]\[ \frac{-7a - 14}{-4} = \frac{7a + 14}{4} \][/tex]
So our inequality now is:
[tex]\[ 3a + 1 > \frac{7a + 14}{4} \][/tex]
### Step 2: Clear the Denominator
To clear the denominator, multiply every term by 4:
[tex]\[ 4(3a + 1) > 7a + 14 \][/tex]
This simplifies to:
[tex]\[ 12a + 4 > 7a + 14 \][/tex]
### Step 3: Move All Terms Involving [tex]\( a \)[/tex] to One Side
Now, move all terms involving [tex]\( a \)[/tex] to one side of the inequality and constants to the other side:
[tex]\[ 12a - 7a > 14 - 4 \][/tex]
This simplifies to:
[tex]\[ 5a > 10 \][/tex]
### Step 4: Solve for [tex]\( a \)[/tex]
Divide both sides of the inequality by 5 to isolate [tex]\( a \)[/tex]:
[tex]\[ a > 2 \][/tex]
### Step 5: Write the Solution
The solution to the inequality [tex]\( 3a + 1 > \frac{-7a - 14}{-4} \)[/tex] is:
[tex]\[ a > 2 \][/tex]
Thus, the result is correctly presented as:
[tex]\[ 2 < a < \infty \][/tex]
It means that [tex]\( a \)[/tex] must be greater than 2. This is the solution to the inequality [tex]\( 3a + 1 > \frac{-7a - 14}{-4} \)[/tex].
