To solve the inequality [tex]\( 3x^2 - 4 \leq 6 - 5x \)[/tex], we need to follow a step-by-step process to isolate [tex]\( x \)[/tex] and determine the solution interval. Here's how we can do it:
### Step 1: Simplify the Inequality
First, let's move all terms to one side of the inequality:
[tex]\[ 3x^2 - 4 \leq 6 - 5x \][/tex]
[tex]\[ 3x^2 - 4 - 6 + 5x \leq 0 \][/tex]
[tex]\[ 3x^2 + 5x - 10 \leq 0 \][/tex]
### Step 2: Solve the Quadratic Equation
We need to solve the corresponding quadratic equation:
[tex]\[ 3x^2 + 5x - 10 = 0 \][/tex]
We use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 3 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -10 \)[/tex].
First, calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 5^2 - 4 \cdot 3 \cdot (-10) \][/tex]
[tex]\[ \Delta = 25 + 120 \][/tex]
[tex]\[ \Delta = 145 \][/tex]
Since the discriminant is positive, we have two real solutions:
[tex]\[ x = \frac{-5 \pm \sqrt{145}}{2 \cdot 3} \][/tex]
[tex]\[ x = \frac{-5 \pm \sqrt{145}}{6} \][/tex]
Calculate the two solutions:
[tex]\[ x_1 = \frac{-5 + \sqrt{145}}{6} \][/tex]
[tex]\[ x_2 = \frac{-5 - \sqrt{145}}{6} \][/tex]
Using a calculator to approximate:
[tex]\[ \sqrt{145} \approx 12.04 \][/tex]
So:
[tex]\[ x_1 = \frac{-5 + 12.04}{6} \approx \frac{7.04}{6} \approx 1.17 \][/tex]
[tex]\[ x_2 = \frac{-5 - 12.04}{6} \approx \frac{-17.04}{6} \approx -2.84 \][/tex]
### Step 3: Determine the Interval
Now we have the critical points [tex]\( x \approx -2.84 \)[/tex] and [tex]\( x \approx 1.17 \)[/tex]. Next, we test intervals around these points to determine where the inequality [tex]\( 3x^2 + 5x - 10 \leq 0 \)[/tex] holds true.
1. Test Interval: [tex]\( x < -2.84 \)[/tex]
2. Test Interval: [tex]\( -2.84 \leq x \leq 1.17 \)[/tex]
3. Test Interval: [tex]\( x > 1.17 \)[/tex]
We need to test values within each interval to see where the inequality holds:
For [tex]\( x < -2.84 \)[/tex]:
Choose [tex]\( x = -3 \)[/tex]:
[tex]\[ 3(-3)^2 + 5(-3) - 10 = 27 - 15 - 10 = 2 \][/tex]
[tex]\( 2 > 0 \)[/tex] (does not satisfy the inequality)
For [tex]\( -2.84 \leq x \leq 1.17 \)[/tex]:
Choose [tex]\( x = 0 \)[/tex]:
[tex]\[ 3(0)^2 + 5(0) - 10 = -10 \][/tex]
[tex]\( -10 \leq 0 \)[/tex] (satisfies the inequality)
For [tex]\( x > 1.17 \)[/tex]:
Choose [tex]\( x = 2 \)[/tex]:
[tex]\[ 3(2)^2 + 5(2) - 10 = 12 + 10 - 10 = 12 \][/tex]
[tex]\( 12 > 0 \)[/tex] (does not satisfy the inequality)
### Step 4: Conclusion
The solutions for the inequality are in the interval:
[tex]\[ -2.84 \leq x \leq 1.17 \][/tex]
Thus, the correct answer is:
[tex]\[ -2.84 \leq x \leq 1.17 \][/tex]
