Math Problem Statement

log 2 ​ (x 2 +x+2)>3

Solution

We are given the inequality:

log2(x2+x+2)>3\log_2(x^2 + x + 2) > 3

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:

x2+x+2>23x^2 + x + 2 > 2^3

Since 23=82^3 = 8, the inequality becomes:

x2+x+2>8x^2 + x + 2 > 8

Step 2: Simplify the inequality

Now, subtract 8 from both sides:

x2+x+28>0x^2 + x + 2 - 8 > 0

This simplifies to:

x2+x6>0x^2 + x - 6 > 0

Step 3: Solve the quadratic inequality

To solve the quadratic inequality, we first solve the corresponding quadratic equation:

x2+x6=0x^2 + x - 6 = 0

We can factor the quadratic expression:

(x2)(x+3)=0(x - 2)(x + 3) = 0

Thus, the solutions to the equation are x=2x = 2 and x=3x = -3. 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 x2+x6x^2 + x - 6 is greater than 0. We check the sign of the factors (x2)(x - 2) and (x+3)(x + 3) in the following intervals:

  • For x<3x < -3: Both x2x - 2 and x+3x + 3 are negative, so the product is positive.
  • For 3<x<2-3 < x < 2: x2x - 2 is negative, and x+3x + 3 is positive, so the product is negative.
  • For x>2x > 2: Both x2x - 2 and x+3x + 3 are positive, so the product is positive.

Thus, the solution to the inequality x2+x6>0x^2 + x - 6 > 0 is:

x<3orx>2x < -3 \quad \text{or} \quad x > 2

Step 5: Check the domain of the logarithmic expression

Since the logarithmic function log2(x2+x+2)\log_2(x^2 + x + 2) is only defined for positive values, we need to ensure that x2+x+2>0x^2 + x + 2 > 0 for all values of xx. The expression x2+x+2x^2 + x + 2 is always positive because the discriminant of the quadratic equation x2+x+2=0x^2 + x + 2 = 0 is negative (12412=71^2 - 4 \cdot 1 \cdot 2 = -7), so it never crosses zero.

Final Answer

The solution to the inequality log2(x2+x+2)>3\log_2(x^2 + x + 2) > 3 is:

x<3orx>2x < -3 \quad \text{or} \quad x > 2

Let me know if you'd like further details or have any questions!

5 Related Questions:

  1. How can I solve a logarithmic inequality with a different base, say base 10?
  2. What is the meaning of the domain restriction in a logarithmic function?
  3. Can you explain the properties of logarithms that help in solving inequalities like this one?
  4. How do you solve logarithmic inequalities involving multiple terms inside the logarithm?
  5. 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