Problem 67 - Algebra and Trigonometry

Question

Solve the inequality. \(\log (x-2)+\log (9-x)<1\)

Step-by-Step Solution

Answer

The solution is \((4, 7)\).

Step 1: Combine Logarithms

Use the property of logarithms that states \( \log a + \log b = \log(ab) \). Here, combine the two logarithms on the left-hand side: \( \log((x-2)(9-x)) < 1 \).

Step 2: Exponentiate to Remove Logarithm

Since the inequality involves a logarithm, exponentiate both sides to eliminate the logarithm: \( (x-2)(9-x) < 10^1 \). This simplifies to \( (x-2)(9-x) < 10 \).

Step 3: Expand the Expression

Expand the expression \((x-2)(9-x)\). This gives: \( x(9-x) - 2(9-x) = 9x - x^2 - 18 + 2x = -x^2 + 11x - 18 \).

Step 4: Set Up Inequality

Write the inequality as: \( -x^2 + 11x - 18 < 10 \). Simplify it to \( -x^2 + 11x - 28 < 0 \).

Step 5: Rearrange the Inequality

Rearrange the terms for easier factoring: \( x^2 - 11x + 28 > 0 \).

Step 6: Factor the Quadratic

Factor the quadratic expression \( x^2 - 11x + 28 \). The factors are \((x-4)(x-7) > 0 \).

Step 7: Find the Solution Interval

The solutions for the equation \((x-4)(x-7) = 0\) are \(x = 4\) and \(x = 7\). Test intervals around these roots to determine where the inequality holds. The intervals are: \((-\infty, 4)\), \((4, 7)\), and \((7, \infty)\). By testing points, we determine the solution interval is \((4, 7)\).

Step 8: Consider Domain of Original Logarithms

Ensure the domain restrictions of the original logarithms are met. The expressions \(x-2\) and \(9-x\) must be greater than zero. This gives the intervals \(x > 2\) and \(x < 9\). The valid intersection with the solution from the quadratic part is \((4, 7)\).

Key Concepts

Logarithmic Inequalities

Logarithmic inequalities involve comparisons within logarithmic expressions. Such inequalities require a solid understanding of logarithm properties.

To handle them, use properties like \( \log a + \log b = \log(ab) \) to simplify the expressions. This step consolidates logs for simpler comparison and solution.

• Watch for domain limitations that constrain the values within the logarithm.
• The base of the logarithm often determines how inequalities are handled when removing the log via exponentiation.

Once simplified, exponentiate both sides to eliminate the logarithm while keeping the inequality intact. This simplification is crucial to solve for 'x' in a clearer form.

Quadratic Inequalities

Quadratic inequalities resemble quadratic equations but involve inequalities instead of equal signs.

They generally take the form \( ax^2 + bx + c < 0 \) or other similar inequalities.

• First, set the quadratic expression to zero to find critical points or roots.
• You can visualize these inequalities by considering the parabola represented by the quadratic expression.

Identify that these points divide the number line into intervals. Testing each interval with values helps determine where the solution to the inequality lies—whether it's inside or outside these regions.

Factoring Quadratics

Factoring is a pivotal skill in solving quadratic inequalities. It simplifies expressions like \( ax^2 + bx + c \) to a more manageable \( (x - p)(x - q) \) form.

Factoring involves recognizing patterns or using techniques like:

• Trial and error to find numbers that satisfy the quadratic's coefficients.
• Using the quadratic formula for more complex expressions.

Once factored, it becomes substantially easier to set each factor to zero and solve for 'x.' These solutions are crucial—they indicate where the expression changes sign, helping determine whether intervals satisfy the original inequality.

Domain Restrictions

Domain restrictions are crucial when solving inequalities, especially with logarithms, since logs are only defined for positive numbers.

These restrictions ensure that solutions remain valid and meaningful.

• For logarithmic expressions like \( \log(x-2) \), ensure \( x - 2 > 0 \) or \( x > 2 \).
• Similarly, \( \log(9 - x) \) necessitates \( 9 - x > 0 \) or \( x < 9 \).

After solving the inequality, always check the domain's conditions. The solution must lie within these feasible intervals to be valid for both inequality and its original functions, resolving to a true solution state like \((4, 7)\), taking domain limits into account.