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Solve the inequality (x - 3)(x - 5)x < 0.

 Jun 23, 2020
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There are three zeros of the inequality:  3, 5, and 0.

None of these will be solutions to the inequality, because there is no equal sign in the problem, 

however, there are four partitions of the real number line:

     x < 0          0 < x < 3          3 < x < 5          x > 5

 

Try a number smaller than 0, say -1, and place this number into the inequality:

     (x - 3)(x - 5)x  <  0     --->     (-1 - 3)(-1 - 5)(-1)  <  0     --->     (-4)(-6)(-1)  <  0     --->     -24  <  0

This is true!, so the whole region  x < 0  is part of the solution set.

 

Next, try a number in the region 0 < x < 3, say 1:

     (x - 3)(x - 5)x  <  0     --->     (1 - 3)(1 - 5)(1)  <  0     --->     (-2)(-4)(1)  <  0     --->     8  <  0

This is not true!, so this whole region is not part of the solution set.

 

Now, try a number in the region 3 < x < 5, say 4:

     (x - 3)(x - 5)x  <  0     --->     (4 - 3)(4 - 5)(4)  <  0     --->     (1)(-1)(4)  <  0     --->     -4  <  0

This is true!, so the whole region  3 < x < 5  is part of the solution set.

 

Next, try a number in the region x > 5, say 8:

     (x - 3)(x - 5)x  <  0     --->     (8 - 3)(8 - 5)(8)  <  0     --->     (5)(3)(8)  <  0     --->     120  <  0

This is not true!, so this whole region is not part of the solution set.

 

The solution set consists of these two regions:  x < 0  and  3 < x < 5.

 Jun 23, 2020

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