Math Problem Statement
Solution
The inequality in the image is:
To solve this inequality, let's go step-by-step.
Step 1: Rewrite the logarithms
We will convert both logarithmic expressions to base 2. We know that:
Since , we can rewrite the second term as:
Step 2: Set up the new inequality
Now, the inequality becomes:
Step 3: Eliminate the negative sign
We can multiply both sides of the inequality by (remember that this reverses the inequality sign):
Or equivalently:
Step 4: Remove the logarithms
Since implies that , we can eliminate the logarithms to get:
Step 5: Solve the inequality
Now, we need to solve the inequality . Let's first multiply both sides by (note that must be positive for this to be valid, so we need to check the domain later):
Now expand the right-hand side:
Simplifying:
Rearrange the inequality:
This is a quadratic inequality. We will solve it by finding the roots of the corresponding equation and determining the intervals.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Inequalities
Logarithm Base Conversion
Quadratic Inequalities
Formulas
log_a(b) = log_c(b) / log_c(a)
log_(1/2)(x) = -log_2(x)
Quadratic equation: ax^2 + bx + c = 0
Theorems
Logarithmic Properties
Quadratic Formula
Suitable Grade Level
High School (Grades 10-12)
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