Solving a Quadratic Inequality with Sign Analysis

0:00Solve the quadratic inequality x squared minus 5 x plus 6 is greater than 0.

0:08Factor: negative 2 and negative 3 multiply to 6 and add to negative 5, giving x minus 2 times x minus 3.

0:18The zeros, where the product is 0, are 2 and 3. They split the number line into three intervals: less than 2, between 2 and 3, and greater than 3.

0:30Sign analysis means testing one value in each interval to find the sign of the product.

0:36Try x equals 0: negative 2 times negative 3 is 6, positive.

0:43Try x equals 2.5: 0.5 times negative 0.5 is negative 0.25, negative.

0:53Try x equals 4: 2 times 1 is 2, positive.

0:59We want greater than 0, so keep the positive intervals: x less than 2 or x greater than 3.

1:07A common mistake is to solve it like an equation: only x equals 2 and x equals 3. In fact the product there is 0, which is not greater than 0, so 2 and 3 are left out.

1:22In interval notation: negative infinity to 2, union 3 to infinity. Union joins the parts, and round brackets leave the ends out.

1:32The graph agrees: the parabola is above the x axis left of 2 and right of 3.

1:40In context: a ball is negative 5 t squared plus 20 t plus 1 metres high after t seconds. When is it above 16 metres?

1:51It is 16 metres high at t equals 1 and 3, and 21 at t equals 2, so it is above 16 metres when 1 is less than t is less than 3.

2:04Find the zeros, test each interval, and keep the intervals with the sign you want.