Math Problem Statement
log 2 (x 2 +x+2)>3
Solution
We are given the inequality:
Step 1: Eliminate the logarithm
To eliminate the logarithm, we need to rewrite the inequality in exponential form. The base of the logarithm is 2, so the inequality becomes:
Since , the inequality becomes:
Step 2: Simplify the inequality
Now, subtract 8 from both sides:
This simplifies to:
Step 3: Solve the quadratic inequality
To solve the quadratic inequality, we first solve the corresponding quadratic equation:
We can factor the quadratic expression:
Thus, the solutions to the equation are and . These are the points where the expression changes sign.
Step 4: Determine the sign of the quadratic expression
Now we need to determine where the quadratic expression is greater than 0. We check the sign of the factors and in the following intervals:
- For : Both and are negative, so the product is positive.
- For : is negative, and is positive, so the product is negative.
- For : Both and are positive, so the product is positive.
Thus, the solution to the inequality is:
Step 5: Check the domain of the logarithmic expression
Since the logarithmic function is only defined for positive values, we need to ensure that for all values of . The expression is always positive because the discriminant of the quadratic equation is negative (), so it never crosses zero.
Final Answer
The solution to the inequality is:
Let me know if you'd like further details or have any questions!
5 Related Questions:
- How can I solve a logarithmic inequality with a different base, say base 10?
- What is the meaning of the domain restriction in a logarithmic function?
- Can you explain the properties of logarithms that help in solving inequalities like this one?
- How do you solve logarithmic inequalities involving multiple terms inside the logarithm?
- What is the significance of the discriminant in solving quadratic inequalities?
Tip: When solving logarithmic inequalities, always check the domain to ensure the expression inside the logarithm is positive.
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Math Problem Analysis
Mathematical Concepts
Logarithmic Inequalities
Quadratic Equations
Exponential Functions
Formulas
log_b(x) = y implies b^y = x
Quadratic equation: ax^2 + bx + c = 0
Theorems
Properties of logarithms
Quadratic factoring method
Suitable Grade Level
Grades 9-12