Solve the Inequality for x 5/(8x+3)< 0
The question asks to find the set of real numbers x that satisfy the inequality 5/(8x+3) < 0. This involves determining the values of x for which the fraction is negative, which means finding out when the numerator is positive and the denominator is negative or vice versa, since a positive divided by a negative or a negative divided by a positive results in a negative number. The inequality does not include the equality condition (it's strictly less than, not less than or equal to), so values that make the denominator zero are not included in the solution set. Solving this problem will likely require identifying the critical value that makes the denominator zero and then using test values or analyzing the number line to find the intervals where the inequality is true.
$\frac{5}{8 x + 3} < 0$
Answer
Solution:
Step 1:
Identify the critical points where the inequality changes by equating the factors to $0$ and solving for $x$.
$8x + 3 = 0$
Step 2:
Isolate $8x$ by subtracting $3$ from both sides.
$8x = -3$
Step 3:
Divide the equation $8x = -3$ by $8$ to find the value of $x$.
Step 3.1:
Divide $8x$ and $-3$ by $8$.
$\frac{8x}{8} = \frac{-3}{8}$
Step 3.2:
Simplify the equation by reducing common factors.
Step 3.2.1:
Eliminate the common factor of $8$.
$\frac{\cancel{8}x}{\cancel{8}} = \frac{-3}{8}$
Step 3.2.2:
Solve for $x$ by dividing by $1$.
$x = \frac{-3}{8}$
Step 4:
Determine the domain of $\frac{5}{8x + 3}$ by finding the values for which the denominator is non-zero.
Step 4.1:
Set the denominator equal to $0$ to find the undefined points.
$8x + 3 = 0$
Step 4.2:
Solve for $x$.
Step 4.2.1:
Subtract $3$ from both sides to isolate $8x$.
$8x = -3$
Step 4.2.2:
Divide by $8$ to find $x$.
$\frac{8x}{8} = \frac{-3}{8}$
Step 4.2.3:
Reduce the equation by canceling out the common factor.
$x = \frac{-3}{8}$
Step 5:
The domain is the set of all $x$ values that keep the expression valid.
$( -\infty, -\frac{3}{8} ) \cup ( -\frac{3}{8}, \infty )$
Step 6:
The solution set includes all intervals where the inequality holds true.
$x < -\frac{3}{8}$
Step 7:
Express the solution in various formats.
Inequality Form: $x < -\frac{3}{8}$ Interval Notation: $(-\infty, -\frac{3}{8})$
Knowledge Notes:
To solve the inequality $\frac{5}{8x+3} < 0$, we need to understand the following concepts:
Critical Points: These are values of $x$ that make the numerator or denominator of a fraction equal to zero. For the inequality $\frac{5}{8x+3} < 0$, the critical point is found by setting the denominator $8x+3$ to zero and solving for $x$.
Test Intervals: Once the critical points are found, the number line is divided into intervals. We test each interval to determine where the inequality holds true.
Domain of a Function: The domain is the set of all possible input values (in this case, $x$ values) that make the function (or expression) defined and real. For a fraction, the domain excludes values that make the denominator zero.
Simplifying Expressions: When solving equations or inequalities, it's important to simplify expressions by canceling common factors and reducing fractions to their simplest form.
Inequality and Interval Notation: The solution to an inequality can be expressed in inequality form (e.g., $x < -\frac{3}{8}$) or interval notation (e.g., $(-\infty, -\frac{3}{8})$), which are both ways to describe the set of numbers that satisfy the inequality.
Multiplication and Division by Negative Numbers: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. However, this rule does not apply to the given problem since we are not multiplying or dividing by a negative number.
By understanding these concepts, we can systematically solve inequalities and express their solutions correctly.
