Exercise
Solve the inequality:
Final Answer
Solution
Move 3 to the other side:
By logarithm definition we get:
Also, we require the phrase inside the log to be greater than zero:
Because otherwise there is no solution to the inequality.
We solve the two inequalities we have received and intersect their results (“and”).
We solve the first inequality:
It is a square inequality. Its roots are 0 and 2.
The coefficient of the square expression (a) is positive, so the parabola (quadratic equation graph) “smiles” (= bowl-shaped). The sign of the inequality means we are looking for the sections the parabola is above the x-axis. The solutions (= zeros = roots) of the quadratic equation are 0 and 2. Hence, the graph goes through the x-axis at these points. Therefore, the solution of the inequality is
We solve the second inequality:
We move everything to one side:
The coefficient of the square expression (a) is positive, so the parabola (quadratic equation graph) “smiles” (= bowl-shaped). The sign of the inequality means we are looking for the sections the parabola is below the x-axis. We find the solutions (= zeros = roots) of the quadratic equation using the quadratic formula. Putting the coefficients in the formula gives us
Hence, we get the solutions:
Because we are looking for the section below the x-axis or on it and the parabola “smiles”, we get that the solution of the inequality is
Finally, we intersect both results ( “and”) and get:
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