Question

Solve the inequality. Then graph the solution set on the real number line. $\frac{3}{5}x-7<8$

Short answer

The solution to the inequality is $\frac{27}{5}\le x\le 3$

Step-by-step solution

Solve the Inequality

Firstly, isolate $x$ by multiplying both sides by the common denominator $4x^2-9$, obtaining: $4x\le 9x-27$. Solving this inequality gives: $5x\ge 27$, from which we have $x\ge \frac{27}{5}$. Remember the quadratic is defined when $x>3$ or $x<-3/4$.

Analyse the Results

So far we've found out that the solution for this inequality is $x\ge \frac{27}{5}$, but we should remember that $x$ must also be such that the denominators $x-3$ and $4x+3$ are non-zero. This gives us three sections to consider: $x<-3/4$, $-3/4\le x<3$, and $x>3$. Considering these restrictions we come to a solution $\frac{27}{5}\le x\le 3$.

Plot on a Number Line

On a real number line, a closed dot is made at $\frac{27}{5}$ and at 3, drawing a line segment between these two dots. Here, both endpoints are included, indicating the inequality is less than or equal to, not just less than.

Key concepts

Real Number Line

The real number line is a fundamental concept in mathematics that helps visualize numbers and their relationships. It is a straight line where each point corresponds to a real number. Small numbers are found on the left, and larger numbers are placed to the right.
An important feature of the real number line is its continuity. There are no gaps between the points, meaning every real number will find its place on the line. When graphing a solution, we can clearly see which numbers are included or excluded.

• If a number is included in a solution, it is marked with a closed dot.
• If a number is not part of the solution, it might be marked with an open dot.
• A line or segment indicates which parts of the number line are solutions.

In our exercise, the solution set $\text{[}\frac{27}{5}3\text{]}$ is represented on the number line by shading the section from $\frac{27}{5}$ to 3, showing that all numbers in this range satisfy the inequality.

Quadratic Inequality

A quadratic inequality involves a polynomial with a degree of two, which makes it more complex than linear inequalities. It typically looks like $a{x}^2+bx+c<0$ or something similar. When solving such inequalities, our aim is to determine where the polynomial is greater or less than zero.
In this exercise, we faced a rational expression that evolved into the quadratic equation $(4x^2-9)\le 0$. Solving it requires factorizations or using the quadratic formula to find critical points. These points help determine sections of the number line where the inequality holds true.
When solving quadratic inequalities:

• Find critical points by setting the equation to zero.
• Determine intervals where the inequality is true or false by testing values.
• Consider edge cases, such as denominators that cannot be equal to zero.

Remember, solving quadratic inequalities often involves checking multiple conditions to ensure the solution is valid.

Solution Set

A solution set is the collection of all values that satisfy an inequality. For this exercise, after solving the inequality and considering the restrictions, the solution set was determined to be $\text{[}\frac{27}{5},3\text{]}$.
Formulating the solution set requires a careful balance of algebraic manipulation and logical reasoning. Ensure that you consider the entire domain $x$ can inhabit, while respecting any restrictions inherent in the problem.

• Start by solving the inequality to isolate the variable $x$.
• Consider restrictions, like making denominators non-zero for rational expressions.
• Combine both results to identify the interval where the inequality holds.

By following these steps, you can accurately describe the breadth of numbers (solutions) that fulfill the inequality, which gets represented on the real number line as shown in the example.