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subtracted (-3)*x from both sides
multiplied both sides by 1/2
subtracted 1 from both sides
multiplied both sided by -1
Factoring left side yields
(x-3)(x-2) >= 0
The values of x for which x-2=0 or x-3=0 are x=2 and x=5. These points divide the coordinate line into three intervals,
(-inf,2], (2,3) and [3, +inf)
We need to check points of which of these three intervals give positive sign for the product (x-3(x-2). We shall choose arbitrary points on each of these intervals to determine the sign; these points are called test points. Lets say 1, 2.5 and 4 will be the test points for intervals (-inf,2], (2,3) and [3, +inf) respectively.
For interval (-inf,2] with test point 1 sign of (x-2)(x-3) is (-)(-) = +
For interval (2,3) with test point 2.5 sign of (x-2(x-3) is (+)(-) = -
For interval [3,+inf)with test point 4 sign of (x-2(x-3) is (+)(+) = +
The pattern of signs suggest that the solution set is
(-inf,2] U [3,+inf)