Questions: Solve the inequality algebraically and write any solution in interval notation. (If there is no solution, enter NO SOLUTION.) x^2+7 x>-12

Transcript text: Solve the inequality algebraically and write any solution in interval notation. (If there is no solution, enter NO SOLUTION.) \[ x^{2}+7 x>-12 \] $\square$

Solution

Solution Steps

Step 1: Rewrite the inequality in standard form

Move all terms to one side of the inequality to set it to zero: $x^{2} + 7x + 12 > 0$

Step 2: Factor the quadratic expression

Factor the quadratic expression $x^{2} + 7x + 12$ : $(x + 3)(x + 4) > 0$

Step 3: Find the critical points

Set each factor equal to zero to find the critical points: $x + 3 = 0 \quad \Rightarrow \quad x = -3$ $x + 4 = 0 \quad \Rightarrow \quad x = -4$

Step 4: Determine the intervals to test

The critical points divide the number line into three intervals:

$x < -4$
$-4 < x < -3$
$x > -3$

Step 5: Test each interval

Choose a test point from each interval and substitute it into the inequality $(x + 3)(x + 4) > 0$ :

For $x < -4$ , choose $x = -5$ : $(-5 + 3)(-5 + 4) = (-2)(-1) = 2 > 0 \quad \text{(True)}$
For $-4 < x < -3$ , choose $x = -3.5$ : $(-3.5 + 3)(-3.5 + 4) = (-0.5)(0.5) = -0.25 < 0 \quad \text{(False)}$
For $x > -3$ , choose $x = 0$ : $(0 + 3)(0 + 4) = (3)(4) = 12 > 0 \quad \text{(True)}$