Solve the Inequality for a 1+5/(a-1)< =7/6
The problem provided presents an inequality that involves a variable 'a' and requires manipulation in order to isolate this variable on one side of the inequality. Specifically, you are asked to solve the inequality:
1 + 5/(a - 1) ≤ 7/6
To solve this inequality, one would typically start by finding a common denominator to combine the terms on the left-hand side of the inequality, then isolate the terms with the variable 'a' on one side, and finally solve for 'a' to find the range of values that satisfy the inequality. The process would involve algebraic manipulations such as addition, subtraction, multiplication, division, and possibly considering the reversal of the inequality sign when multiplying or dividing by negative numbers, depending on the specific steps required to isolate 'a'.
$1 + \frac{5}{a - 1} \leq \frac{7}{6}$
Answer
Solution:
Step 1
Eliminate $\frac{7}{6}$ from both sides of the inequality to get $1 + \frac{5}{a - 1} - \frac{7}{6} \leq 0$.
Step 2
Begin simplifying the expression $1 + \frac{5}{a - 1} - \frac{7}{6}$.
Step 2.1
Express $1$ as $\frac{a - 1}{a - 1}$ to obtain a common denominator.
Step 2.2
Combine the terms in the numerator to get $\frac{a - 1 + 5}{a - 1} - \frac{7}{6} \leq 0$.
Step 2.3
Simplify the numerator to $\frac{a + 4}{a - 1} - \frac{7}{6} \leq 0$.
Step 2.4
Multiply $\frac{a + 4}{a - 1}$ by $\frac{6}{6}$ to match the denominators.
Step 2.5
Multiply $-\frac{7}{6}$ by $\frac{a - 1}{a - 1}$ to match the denominators.
Step 2.6
Adjust both fractions to have the common denominator $(a - 1) \cdot 6$.
Step 2.6.1
Multiply $\frac{a + 4}{a - 1}$ by $\frac{6}{6}$.
Step 2.6.2
Multiply $\frac{7}{6}$ by $\frac{a - 1}{a - 1}$.
Step 2.6.3
Reorder the factors to get $\frac{(a + 4) \cdot 6}{6(a - 1)} - \frac{7(a - 1)}{6(a - 1)} \leq 0$.
Step 2.7
Combine the numerators over the common denominator to get $\frac{(a + 4) \cdot 6 - 7(a - 1)}{6(a - 1)} \leq 0$.
Step 2.8
Simplify the numerator.
Step 2.8.1
Distribute to get $\frac{6a + 24 - 7(a - 1)}{6(a - 1)} \leq 0$.
Step 2.8.2
Rearrange to $\frac{6a + 24 - 7a + 7}{6(a - 1)} \leq 0$.
Step 2.8.3
Combine like terms to get $\frac{-a + 31}{6(a - 1)} \leq 0$.
Step 2.8.4
Factor out $-1$ to get $-\frac{a - 31}{6(a - 1)} \leq 0$.
Step 3
Identify the critical points by setting the numerator and denominator to $0$.
Step 4
Solve $a - 31 = 0$ to find $a = 31$.
Step 5
Solve $a - 1 = 0$ to find $a = 1$.
Step 6
Determine the intervals where the expression changes sign.
Step 7
Combine the critical points to get $a = 31, 1$.
Step 8
Determine the domain of $-\frac{a - 31}{6(a - 1)}$.
Step 8.1
Set the denominator equal to $0$ to find the undefined points.
Step 8.2
Solve for $a$ to find $a = 1$.
Step 8.3
The domain is all $a$ except $a = 1$.
Step 9
Create test intervals around the critical points.
Step 10
Test values from each interval in the original inequality.
Step 10.1
Test a value from $a < 1$.
Step 10.2
Test a value from $1 < a < 31$.
Step 10.3
Test a value from $a > 31$.
Step 10.4
Determine which intervals satisfy the inequality.
Step 11
The solution includes all intervals that make the inequality true.
Step 12
Present the solution in inequality and interval notation.
Step 13
Finalize the solution.
Knowledge Notes:
To solve an inequality involving fractions, one must first clear the fractions by finding a common denominator. The inequality can then be simplified by combining like terms and factoring when necessary. Critical points are found by setting the numerator and denominator equal to zero, as these are the values where the expression can change sign. The domain must exclude values that make the denominator zero, as they are undefined. Test intervals are created around the critical points, and test values from each interval are plugged into the original inequality to determine which intervals satisfy the inequality. The solution is presented in both inequality and interval notation.
