(x+19)(x-7)(x+17)>0=-19<x<-17 or x>7
Can someone please look at this equation and let me know if I did it correctly.
If x>7, all factors are positive and the product is positive. So that is OK
If -19<x<-17, two factors are negative and x+19 is positive, so the product is positive. Also OK
There are no other regions where the function is positive. It is a cubic equation with three roots. Between the secoind and third roots, it is negative.
Thank you
Yes, you have determined the correct solution to the inequality (x+19)(x-7)(x+17) > 0. The equation is already factored, so we just need to determine the intervals where the expression is positive (greater than zero).
To solve an inequality, we first find the critical points where the expression equals zero. In this case, we have three critical points: x = -19, x = 7, and x = -17. These are the values that make each factor equal to zero.
Next, we create sign charts to determine the sign of the expression within each interval. We start by plotting the critical points on a number line and picking a test value within each interval.
- For x < -19, we can choose x = -20 as a test value. Plugging it into the inequality, we have (-20+19)(-20-7)(-20+17) > 0, which simplifies to (-1)(-27)(-3) > 0. Since there are two negative factors, the expression is positive. Therefore, the inequality holds for x < -19.
- For -19 < x < -17, we can choose x = -18 as a test value. Plugging it into the inequality, we have (-18+19)(-18-7)(-18+17) > 0, which simplifies to (1)(-25)(-1) > 0. Since there is one positive and two negative factors, the expression is negative. Therefore, the inequality does not hold for -19 < x < -17.
- For -17 < x < 7, we can choose x = 0 as a test value. Plugging it into the inequality, we have (0+19)(0-7)(0+17) > 0, which simplifies to (19)(-7)(17) > 0. Since there is one positive and two negative factors, the expression is negative. Therefore, the inequality does not hold for -17 < x < 7.
- For x > 7, we can choose x = 8 as a test value. Plugging it into the inequality, we have (8+19)(8-7)(8+17) > 0, which simplifies to (27)(1)(25) > 0. Since all three factors are positive, the expression is positive. Therefore, the inequality holds for x > 7.
Putting it all together, we have the solution: -19 < x < -17 or x > 7.