Question

In Problems \(7-22,\) solve each inequality. 13\. \(x^{2}+x>12\)

Short answer

x in (-∞, -4) or (3, ∞)

Step-by-step solution

Rewrite the inequality in standard form

Start by moving all terms to one side of the inequality to set it to zero: \[ x^2 + x - 12 > 0 \]

Factor the quadratic expression

Factor the quadratic expression \( x^2 + x - 12 \): \[ x^2 + x - 12 = (x + 4)(x - 3) \]

Determine the critical points

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

Test intervals around the critical points

Divide the number line into intervals based on the critical points (-∞, -4), (-4, 3), (3, ∞) and test a value from each interval in the inequality \( (x + 4)(x - 3) > 0 \): For \( x = -5 \): \[ (x + 4)(x - 3) = (-1)(-8) = 8 > 0 \] (True) For \( x = 0 \): \[ (x + 4)(x - 3) = 4(-3) = -12 < 0 \] (False) For \( x = 4 \): \[ (x + 4)(x - 3) = 8(1) = 8 > 0 \] (True)

Conclusion about the solution

Since the inequality is satisfied in the intervals \((-∞, -4)\) and \((3, ∞)\), the solution is: \[ x \text{ in } (-∞, -4) \text{ or } (3, ∞) \]

Key concepts

Factoring Quadratics

Understanding how to factor quadratic expressions is essential in solving quadratic inequalities. Start with the standard form of a quadratic equation, which is typically ax^2 + bx + c = 0. In our problem, we have x^2 + x - 12.
To factor this, you need to find two numbers that multiply together to give you the constant term (-12) and add up to give you the coefficient of the middle term (1). Those numbers in our case are 4 and -3 because:

• Multiplying: 4 * (-3) = -12
• Adding: 4 + (-3) = 1

This means we can write the expression as (x + 4)(x - 3).This factored form will help us in determining the critical points, which are essential for solving the inequality.

Critical Points

Critical points are the values of x that make the expression equal to zero. These points essentially break down the number line into segments we can test for the inequality. To identify these points, set each factor of the quadratic to zero and solve.

• For (x + 4) = 0, x = -4
• For (x - 3) = 0, x = 3

Thus, x = -4 and x = 3 are our critical points. These points help us to construct the intervals that we need to test.

Interval Testing

Testing intervals involves dividing the number line into sections based on the critical points and then checking the inequality within those sections. For our problem, the critical points were x = -4 and x = 3, dividing the number line into three intervals:

• (-∞, -4)
• (-4, 3)
• (3, ∞)

Now, we pick a random point within each interval and plug it into the inequality (x + 4)(x - 3) > 0 to see if the inequality holds:

• For x = -5 (in interval (-∞, -4)): (x + 4)(x - 3) = (-1)(-8) = 8 > 0 (True)
• For x = 0 (in interval (-4, 3)): (x + 4)(x - 3) = 4(-3) = -12 < 0 (False)
• For x = 4 (in interval (3, ∞)): (x + 4)(x - 3) = 8(1) = 8 > 0 (True)

This interval testing helps us to determine where the inequality is true and where it is false.

Inequality Solutions

The solution to the inequality tells us which intervals satisfy the inequality. From the interval testing, we found that the inequality holds true for intervals (-∞, -4) and (3, ∞). Hence, this means:

• For x in (-∞, -4), the inequality (x + 4)(x - 3) > 0 is satisfied.
• For x in (3, ∞), the inequality (x + 4)(x - 3) > 0 is also satisfied.

Thus, the complete solution to the quadratic inequality x^2 + x - 12 > 0 is: x is in (-∞, -4) or (3, ∞). This means that any value of xtuated within these intervals will make the original inequality true.