Answer :
To solve the inequality [tex]\(x^2 + 4x > 77\)[/tex], we will follow a step-by-step process to determine the values of [tex]\(x\)[/tex] that satisfy this condition.
### Step 1: Move all terms to one side of the inequality
First, we rewrite the inequality by moving 77 to the left side:
[tex]\[ x^2 + 4x - 77 > 0 \][/tex]
### Step 2: Solve the corresponding equation
Solve the equation [tex]\(x^2 + 4x - 77 = 0\)[/tex] to find the critical points where the expression equals zero. These points will help us determine the intervals to test for the inequality.
#### Factor the quadratic equation
We need to factor [tex]\(x^2 + 4x - 77\)[/tex]. To do this, we look for two numbers that multiply to [tex]\(-77\)[/tex] and add up to [tex]\(4\)[/tex]. After carefully considering the factors of [tex]\(-77\)[/tex], we find:
[tex]\[ (x + 11)(x - 7) = 0 \][/tex]
This gives us the critical points:
[tex]\[ x = -11 \quad \text{and} \quad x = 7 \][/tex]
### Step 3: Test intervals around the critical points
To determine where [tex]\(x^2 + 4x - 77 > 0\)[/tex], we test the intervals defined by the critical points. The critical points divide the number line into three intervals: [tex]\((-\infty, -11)\)[/tex], [tex]\((-11, 7)\)[/tex], and [tex]\((7, \infty)\)[/tex].
1. Interval [tex]\((-\infty, -11)\)[/tex]: Choose [tex]\(x = -12\)[/tex]
[tex]\[ (-12)^2 + 4(-12) - 77 = 144 - 48 - 77 = 19 \][/tex]
Since [tex]\(19 > 0\)[/tex], the inequality [tex]\(x^2 + 4x - 77 > 0\)[/tex] holds in this interval.
2. Interval [tex]\((-11, 7)\)[/tex]: Choose [tex]\(x = 0\)[/tex]
[tex]\[ 0^2 + 4(0) - 77 = -77 \][/tex]
Since [tex]\(-77 < 0\)[/tex], the inequality [tex]\(x^2 + 4x - 77 > 0\)[/tex] does not hold in this interval.
3. Interval [tex]\((7, \infty)\)[/tex]: Choose [tex]\(x = 8\)[/tex]
[tex]\[ 8^2 + 4(8) - 77 = 64 + 32 - 77 = 19 \][/tex]
Since [tex]\(19 > 0\)[/tex], the inequality [tex]\(x^2 + 4x - 77 > 0\)[/tex] holds in this interval.
### Conclusion
The inequality [tex]\(x^2 + 4x - 77 > 0\)[/tex] is satisfied when:
[tex]\[ x < -11 \quad \text{or} \quad x > 7 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{x < -11 \text{ or } x > 7} \][/tex]
### Step 1: Move all terms to one side of the inequality
First, we rewrite the inequality by moving 77 to the left side:
[tex]\[ x^2 + 4x - 77 > 0 \][/tex]
### Step 2: Solve the corresponding equation
Solve the equation [tex]\(x^2 + 4x - 77 = 0\)[/tex] to find the critical points where the expression equals zero. These points will help us determine the intervals to test for the inequality.
#### Factor the quadratic equation
We need to factor [tex]\(x^2 + 4x - 77\)[/tex]. To do this, we look for two numbers that multiply to [tex]\(-77\)[/tex] and add up to [tex]\(4\)[/tex]. After carefully considering the factors of [tex]\(-77\)[/tex], we find:
[tex]\[ (x + 11)(x - 7) = 0 \][/tex]
This gives us the critical points:
[tex]\[ x = -11 \quad \text{and} \quad x = 7 \][/tex]
### Step 3: Test intervals around the critical points
To determine where [tex]\(x^2 + 4x - 77 > 0\)[/tex], we test the intervals defined by the critical points. The critical points divide the number line into three intervals: [tex]\((-\infty, -11)\)[/tex], [tex]\((-11, 7)\)[/tex], and [tex]\((7, \infty)\)[/tex].
1. Interval [tex]\((-\infty, -11)\)[/tex]: Choose [tex]\(x = -12\)[/tex]
[tex]\[ (-12)^2 + 4(-12) - 77 = 144 - 48 - 77 = 19 \][/tex]
Since [tex]\(19 > 0\)[/tex], the inequality [tex]\(x^2 + 4x - 77 > 0\)[/tex] holds in this interval.
2. Interval [tex]\((-11, 7)\)[/tex]: Choose [tex]\(x = 0\)[/tex]
[tex]\[ 0^2 + 4(0) - 77 = -77 \][/tex]
Since [tex]\(-77 < 0\)[/tex], the inequality [tex]\(x^2 + 4x - 77 > 0\)[/tex] does not hold in this interval.
3. Interval [tex]\((7, \infty)\)[/tex]: Choose [tex]\(x = 8\)[/tex]
[tex]\[ 8^2 + 4(8) - 77 = 64 + 32 - 77 = 19 \][/tex]
Since [tex]\(19 > 0\)[/tex], the inequality [tex]\(x^2 + 4x - 77 > 0\)[/tex] holds in this interval.
### Conclusion
The inequality [tex]\(x^2 + 4x - 77 > 0\)[/tex] is satisfied when:
[tex]\[ x < -11 \quad \text{or} \quad x > 7 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{x < -11 \text{ or } x > 7} \][/tex]
