Solve the Inequality for x 5/(8x+3)< 0

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.

$(-\mathrm{\infty },-\frac{3}{8})\cup (-\frac{3}{8},\mathrm{\infty })$

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: $(-\mathrm{\infty },-\frac{3}{8})$

Knowledge Notes:

To solve the inequality $\frac{5}{8x+3}<0$, we need to understand the following concepts:

1. 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 $x$.

2. 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.

3. 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.

4. Simplifying Expressions: When solving equations or inequalities, it's important to simplify expressions by canceling common factors and reducing fractions to their simplest form.

5. 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., $(-\mathrm{\infty },-\frac{3}{8})$), which are both ways to describe the set of numbers that satisfy the inequality.

6. 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.