solve the inequality algebraically
7x-6 less than or equal to 3x^2
please show work
3x^2>7x-6
3 x^2 - 7 x + 6 > 0
parabola upright(holds water)
zeros at 7/6 +/- (1/6)sqrt(49-36)
7/6 +/ - (1/6)sqrt(13)
so
x< 7/6 - (1/6) sqrt 13
or
x> 7/6 + (1/6) sqrt 13
good, except it's ≤, not <
To solve the inequality algebraically, we will follow these steps:
Step 1: Rewrite the inequality in standard quadratic form.
Step 2: Set the quadratic expression equal to zero and factor it.
Step 3: Determine the critical points.
Step 4: Test each interval to determine the sign of the inequality.
Step 5: Write the final solution.
Let's go through each step in detail:
Step 1: Rewrite the inequality in standard quadratic form.
The given inequality is: 7x - 6 ≤ 3x^2
Step 2: Set the quadratic expression equal to zero and factor it.
We need to get the quadratic expression on one side of the equation. To do that, we subtract 7x from both sides:
-6 ≤ 3x^2 - 7x
Next, we set the equation equal to zero by subtracting -6 from both sides:
0 ≤ 3x^2 - 7x + 6
Step 3: Determine the critical points.
To factor the quadratic expression 3x^2 - 7x + 6, we find two numbers whose sum is -7 and whose product is 6. The factors are -1 and -6:
0 ≤ (3x - 2)(x - 3)
Now, set each factor equal to zero:
3x - 2 = 0, which gives x = 2/3,
x - 3 = 0, which gives x = 3.
Step 4: Test each interval to determine the sign of the inequality.
To determine the intervals where the inequality is true or false, test the sign of the expression (3x - 2)(x - 3) in each interval:
Interval 1: x < 2/3
Choose a value less than 2/3, like x = 0. Substitute it into the expression:
(3 * 0 - 2)(0 - 3) = (-2)(-3) = 6 > 0
Interval 2: 2/3 < x < 3
Choose a value within the interval, like x = 1. Substitute it into the expression:
(3 * 1 - 2)(1 - 3) = (1)(-2) = -2 < 0
Interval 3: x > 3
Choose a value greater than 3, like x = 4. Substitute it into the expression:
(3 * 4 - 2)(4 - 3) = (10)(1) = 10 > 0
Step 5: Write the final solution.
The solution to the inequality 7x - 6 ≤ 3x^2 is:
x ≤ 2/3 or x > 3
Remember to always double-check your work and include any other conditions or restrictions specified in the original problem.