You can put this solution on YOUR website! a) Solve
This can be solved in a manner similar to solving the related quadratic equation. In other words, get one side equal to zero and then factor the other side. So we'll start by subtracting 3 from both sides:
Then factor:
What we have now is a product (multiplication) which is greater than 0. In other words, we have a product that is positive. And how do we multiply two numbers and end up with a positive result? It should be clear that the two numbers must be both positive or both negative. So we just state this in the form of appropriate inequalities:
(2x - 3 > 0 and x + 1 > 0) or (2x - 3 < 0 and x + 1 < 0)
The first two state that the factors are positive and the last two state that the factors are negative and since either of these will produce a positive result we separate the two pairs of inequalities by an "or".
Now we just solve these four inequalities:
(2x > 3 and x > -1) or (2x < 3 and x < -1)
(x > 3/2 and x > -1) or (x < 3/2 and x < -1)
The left pair of inequalities says that x must be greater than 3/2 and it must be greater than -1. With some thought it should be clear that only numbers greater than 3/2 would fit both. So we will replace the pair with just x > 3/2.
The right pair of inequalities says that x must be less than 3/2 and it must be less than -1. With some thought it should be clear that only numbers less than -1 would fit both. So we will replace the pair with just x < -1.:
x > 3/2 or x < -1
And this is our solution. Any number that is greater than 3/2 or less than -1 will work.
b) Simplify
To simplify logarithms we often use the following properties:
Any of these propoerties may be used (in either direction) and in this problem we will use two of the three.
On the first term we will use the third property (from right to left) to "move" the coefficient (the "2") from in front of the log into the argument:
Now, since we have subtraction of logarithms of the same base, we can use the second property (from right to left) to combine the logarithms:
Now we can simplify:
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