Chapter 10: Q. 3 (page 700)

For what numbers x is $6 - x \geq x^{2}$ ?

Short Answer

Expert verified

The solution set is $[- 3, 2]$

Step by step solution

Step 1. Given information

The given inequality is $6 - x \geq x^{2}$

Step 2. Calculation

Rewrite the given inequality,

$x^{2} + x - 6 \leq 0$

Find the y-intercept of the function $f (x) = x^{2} + x - 6$ by substituting $x = 0$

$f (0) = {(0)}^{2} + 0 - 6 = - 6$

Factorize the equation as follows,

$x^{2} + x - 6 = 0 (x - 2) (x + 3) = 0 x = 2, - 3$

Thus, y-intercept is $- 6$ and x-intercepts are $2, - 3$

Step 3. Calculation

Compare the given function with the standard form $a x^{2} + b x + c = 0$ ,

$a = 1, b = 1, c = - 6$

The x-coordinate of the vertex is as follows,

$- \frac{b}{2 a} = - \frac{1}{2}$

Find the y-coordinate of the vertex,

$f (- \frac{1}{2}) = {(- \frac{1}{2})}^{2} - (\frac{1}{2}) + 6 = \frac{1}{4} - \frac{1}{2} + 6 = \frac{1 - 2 - 24}{4} = - \frac{25}{4}$

Graph the function with vertex $(- \frac{1}{2}, - \frac{25}{4})$ and the points $(- 3, 0), (0, - 6) a n d (2, 0)$

Thus, the solution set is $\{x | - 3 \leq x \leq 2\}$

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For what numbers x is $6 - x \geq x^{2}$ ?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

Step 3. Calculation

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For what numbers x is6-x≥x2?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculation

Step 3. Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

For what numbers x is $6 - x \geq x^{2}$ ?