Math

QuestionSolve the inequality 2x83x22x - 8 \geq -3x^2 and express the solution in interval notation.

Studdy Solution

STEP 1

Assumptions1. We are solving the inequality x -8 \geq -3x^. . The solution set should be expressed in interval notation.
3. We are allowed to use a graphing utility to verify the solution.

STEP 2

First, we need to rearrange the inequality to a standard quadratic inequality form. This is done by adding x2x^2 to both sides of the inequality.
x2+2x80x^2 +2x -8 \geq0

STEP 3

Next, we need to factor the quadratic expression on the left side of the inequality.To factor, we need to find two numbers that multiply to 24-24 (the product of 33 and 8-8) and add to 22 (the coefficient of xx). These numbers are $$ and $-6$.
So, we can write the quadratic expression as follows(3x)(x+2)0(3x -)(x +2) \geq0

STEP 4

Now, we need to find the roots of the quadratic equation 3x4=03x -4 =0 and x+2=0x +2 =0.
For 3x4=03x -4 =0, we add 44 to both sides and then divide by 33 to solve for xxx=4/3x =4/3For x+2=0x +2 =0, we subtract 22 from both sides to solve for xxx=2x = -2

STEP 5

We now have the roots of the inequality, x=4/3x =4/3 and x=2x = -2. These are the points where the inequality equals 00.We can use these roots to divide the number line into three intervals (,2)(-\infty, -2), (2,4/3)(-2,4/3), and (4/3,)(4/3, \infty).

STEP 6

Next, we need to test the sign of the inequality in each interval.Choose a test point in each interval and substitute it into the inequality. If the result is positive, the inequality is true for that interval. If the result is negative, the inequality is false for that interval.
For the interval (,2)(-\infty, -2), choose x=3x = -3(3(3)4)((3)+2)=(13)(1)=13>0(3(-3) -4)((-3) +2) = (-13)(-1) =13 >0So, the inequality is true for (,2)(-\infty, -2).

STEP 7

For the interval (2,4/3)(-2,4/3), choose x=0x =0(3(0)4)((0)+2)=(4)(2)=<0(3(0) -4)((0) +2) = (-4)(2) = - <0So, the inequality is false for (2,4/3)(-2,4/3).

STEP 8

For the interval (4/3,)(4/3, \infty), choose x=2x =2(3(2)4)((2)+2)=(2)(4)=8>0(3(2) -4)((2) +2) = (2)(4) =8 >0So, the inequality is true for (4/3,)(4/3, \infty).

STEP 9

From the test points, we can see that the inequality is true for the intervals (,2)(-\infty, -2) and (4/3,)(4/3, \infty).However, we need to include the roots 2-2 and 4/34/3 in our solution set because the original inequality is \geq, not just >>.
So, the solution set in interval notation is (,2][4/3,)(-\infty, -2] \cup [4/3, \infty).

STEP 10

To verify this solution, we can graph the inequality 2x83x22x -8 \geq -3x^2 using a graphing utility.
The graph should show that the parabola y=3x2+2x8y =3x^2 +2x -8 is above or on the x-axis (i.e., y0y \geq0) for x2x \leq -2 and x4/3x \geq4/3.
The solution set is (,2][4/3,)(-\infty, -2] \cup [4/3, \infty).

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