Problem 47 - Beginning and Intermediate Algebra

Question

Solve each inequality, and graph the solution set. $$ \frac{x-8}{x-4}<3 $$

Step-by-Step Solution

Answer

The solution set is (-∞, 2) ∪ (4, ∞).

Step 1: Move the Constant Term

Begin by subtracting 3 from both sides of the inequality. \[\frac{x-8}{x-4} - 3 < 0\]

Step 2: Combine the Fractions

Express 3 with a common denominator and combine the fractions: \[\frac{x-8 - 3(x-4)}{x-4} < 0\]This simplifies to: \[\frac{x-8 - 3x + 12}{x-4} < 0\]

Step 3: Simplify the Fraction

Combine like terms in the numerator: \[\frac{-2x + 4}{x-4} < 0\]This can be further simplified to: \[\frac{-2(x-2)}{x-4} < 0\]

Step 4: Analyze Critical Points

Find the critical points where the expression equals zero or is undefined. The critical points are where the numerator and denominator are zero: \(x = 2\) (numerator is zero) \(x = 4\) (denominator is zero, undefined)These points divide the number line into intervals to test the inequality.

Step 5: Test Intervals

Choose test points from each interval (-∞, 2), (2, 4), and (4, ∞) to determine where the inequality holds:Let’s test the point \(x = 0\) for the interval (-∞, 2): \[\frac{-2(0-2)}{0-4} = \frac{4}{-4} = -1 < 0 \quad (True)\]Test the point \(x = 3\) for the interval (2, 4): \[\frac{-2(3-2)}{3-4} = \frac{-2}{-1} = 2 > 0 \quad (False)\]Test the point \(x = 5\) for the interval (4, ∞): \[\frac{-2(5-2)}{5-4} = \frac{-6}{1} = -6 < 0 \quad (True)\]

Step 6: Write the Solution

Combine the intervals where the inequality is satisfied: (-∞, 2) ∪ (4, ∞). The solution set excludes the points where the function is undefined or equal to zero.

Step 7: Graph the Solution

Graph the solution set on a number line. Use open circles at the critical points x=2 and x=4 (since they are not included in the solution set) and shade the regions (-∞, 2) and (4, ∞) to represent the intervals where the inequality is satisfied.

Key Concepts

inequality graphing

Graphing an inequality helps us visually understand the range of values that satisfy the inequality.
In this exercise, we are dealing with the expression \((x-8)/(x-4) < 3\).
After simplifying, we find the inequality in terms of critical points.
To graph the solution set, critical points x = 2 and x = 4 are placed on a number line.
These points divide the line into intervals: (-∞, 2), (2, 4), and (4, ∞).
We then test each interval to see if values in those ranges satisfy the inequality.
Shaded regions and open circles (indicating points not included in the solution set) represent where the inequality holds true.
With a correct graph, understanding becomes easier, showing clearly where solutions lie.

rational inequalities

Rational inequalities involve expressions with fractions where the numerator and/or denominator are polynomials.
For the inequality \((x-8)/(x-4) < 3\), we first transform it into a standard form \((x-8)/(x-4) - 3 < 0\).
Combining fractions into one helps to find common grounds to analyze better.
Thus, we simplify to get \((-2(x-2))/(x-4) < 0\).
We then identify critical points, where our numerator or denominator equals zero.
Numerator zero at x = 2, denominator undefined at x = 4.
These critical points help divide the number line and test intervals for where the inequality holds true.
Understanding rational inequalities requires practice to handle complex expressions efficiently.

critical points

Critical points in inequalities are values where the expression equals zero or is undefined.
For example, in \((-2(x-2))/(x-4) < 0\), x = 2 makes the numerator zero and x = 4 makes the denominator undefined.
These points split the number line into intervals which are tested to see if they meet the inequality's conditions.
Testing involves substituting values from each interval back into the simplified inequality.
If the result is true, that interval is part of the solution set; otherwise, it is not.
Critical points and the test intervals between can significantly aid in discovering where inequalities hold true.