solve for the follow inequality. write your solution in interval notation.
x^3-49x¡Ü0
To solve the inequality x^3 - 49x ≤ 0, we need to find the values of x that satisfy this inequality. Here's how you can get the solution:
Step 1: Factor the expression x^3 - 49x.
x^3 - 49x = x(x^2 - 49) = x(x - 7)(x + 7)
Step 2: Set each factor equal to zero and solve for x.
x = 0 (from x = 0)
x - 7 = 0 (from x - 7 = 0)
x = 7
x + 7 = 0 (from x + 7 = 0)
x = -7
Step 3: Use a number line to determine the intervals that make the inequality true.
On the number line, plot the three values: -7, 0, and 7.
From left to right, the intervals are:
1. x < -7
2. -7 < x < 0
3. 0 < x < 7
4. x > 7
Step 4: Determine the sign of x(x - 7)(x + 7) within each interval.
For the first interval, select a value less than -7, for example, -10. Evaluating the expression x(x - 7)(x + 7) at x = -10:
(-10)(-10 - 7)(-10 + 7) = (-10)(-17)(-3) = -510
Since -510 < 0, the expression is negative in this interval.
For the second interval, select a value between -7 and 0, for example, -5. Evaluating the expression x(x - 7)(x + 7) at x = -5:
(-5)(-5 - 7)(-5 + 7) = (-5)(-12)(2) = 120
Since 120 > 0, the expression is positive in this interval.
For the third interval, select a value between 0 and 7, for example, 5. Evaluating the expression x(x - 7)(x + 7) at x = 5:
(5)(5 - 7)(5 + 7) = (5)(-2)(12) = -120
Since -120 < 0, the expression is negative in this interval.
For the fourth interval, select a value greater than 7, for example, 10. Evaluating the expression x(x - 7)(x + 7) at x = 10:
(10)(10 - 7)(10 + 7) = (10)(3)(17) = 510
Since 510 > 0, the expression is positive in this interval.
Step 5: Determine the solution using interval notation.
Based on the sign analysis, we can write the solution in interval notation as:
(-∞, -7) ∪ (0, 7)
This means that x can take any value less than -7 or any value between 0 and 7, excluding -7 and 0.