Chapter 3: Q. 13 (page 171)

In problems 7-22, solve the inequality.
$x^{2} + x > 12$

Short Answer

Expert verified

The solution set is $(- \infty, - 4) o r (3, \infty)$ .

Step by step solution

Step 1. Given Information

The given inequality is $x^{2} + x > 12$ .

Step 2. Find the intercepts and vertex

Rewrite the given inequality.

$x^{2} + x - 12 > 0$

The value of $f (0) = - 12$ .
So, the y-intercept is $(0, - 12)$ .
Factor the equation.

role="math" localid="1646467056241" $\begin{array}{rcl} x^{2} + x - 12 & = & 0 \\ x^{2} + 4 x - 3 x - 12 & = & 0 \\ x (x + 4) - 3 (x + 4) & = & 0 \\ (x - 3) (x + 4) & = & 0 \end{array}$

Equate the factors to 0.

$\begin{array}{rcl} x - 3 & = & 0 \\ x & = & 3 \\ x + 4 & = & 0 \\ x & = & - 4 \end{array}$

So, the x-intercepts are $(3, 0), (- 4, 0)$ .
The vertex of the parabola exists at $\frac{- b}{2 a} = \frac{- 1}{2 (1)} = \frac{- 1}{2}$ .
Find the value of the function at $x = - \frac{1}{2}$ .

$\begin{array}{rcl} f (- \frac{1}{2}) & = & {(- \frac{1}{2})}^{2} + (- \frac{1}{2}) - 12 \\ = & \frac{1}{4} - \frac{1}{2} - 12 \\ = & \frac{1 - 2 - 48}{4} \\ = & \frac{- 49}{4} \end{array}$

So, the vertex is $(- \frac{1}{2}, \begin{array}{rcl} \frac{- 49}{4} \end{array})$ .

Step 3. Plot the graph

Plot the parabola on a graph using the vertex and intercepts calculated in the previous step.

Step 4. Find the solution set

From the graph, it is observed that for x belonging to $(- \infty, - 4) o r (3, \infty)$ , $y > 0$ .
So, the solution set of the given inequality is $(- \infty, - 4) o r (3, \infty)$ .

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In problems 7-22, solve the inequality.
$x^{2} + x > 12$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the intercepts and vertex

Step 3. Plot the graph

Step 4. Find the solution set

One App. One Place for Learning.

Most popular questions from this chapter

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In problems 7-22, solve the inequality. x2+x>12

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Find the intercepts and vertex

Step 3. Plot the graph

Step 4. Find the solution set

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Geometry

Calculus

Statistics

Theoretical and Mathematical Physics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In problems 7-22, solve the inequality.
$x^{2} + x > 12$