Solving The Inequality: X² - 2 < (7/2)x

Dec 3, 2025 by Alex Johnson 40 views

Hey there, math enthusiasts! Today, we're diving into the world of inequalities, specifically tackling the problem: $x^2 - 2 < rac{7}{2}x$ . This might look intimidating at first, but don't worry! We'll break it down step-by-step, making sure you understand the process and the solution.

Understanding the Inequality

Let's start by understanding what this inequality actually means. The expression $x^2 - 2 < rac{7}{2}x$ is a quadratic inequality. Quadratic inequalities involve a quadratic expression (in this case, $x^2 - 2$ ) compared to another expression (here, $rac{7}{2}x$ ). Solving such inequalities means finding the range(s) of values for x that make the inequality true.

The key to solving these types of problems lies in manipulating the inequality to get a clearer picture of what's going on. Our first goal is to rearrange the inequality so that we have a zero on one side. This will allow us to analyze the quadratic expression more easily. To begin, we need to remove the fraction to make our calculations easier and more precise. Fractions can sometimes make things look more complicated than they are, so let's get rid of it by multiplying all terms by 2. This will maintain the balance of our equation while simplifying the terms. By multiplying both sides by 2, we transform the inequality into a more manageable form, setting the stage for the next steps in solving for x. Now, our inequality looks much cleaner, making it easier to work with and understand.

Rearranging the terms is another critical step. We want to ensure that all terms are on one side of the inequality, leaving zero on the other side. This is essential for identifying the critical points where the expression equals zero, which are vital for determining the solution intervals. By bringing all terms to one side, we create a standard quadratic form that is easier to factor and solve. This process not only simplifies the inequality but also makes it clearer to visualize the quadratic expression’s behavior. Remember, the goal is to transform the inequality into a form that is simple and straightforward to analyze. With this arrangement, we can now focus on factoring or using the quadratic formula to find the roots, which are essential for solving the inequality. This step is fundamental in transitioning the problem from an abstract inequality to a concrete equation that can be solved using standard algebraic techniques.

Step-by-Step Solution

Multiply both sides by 2: To get rid of the fraction, let's multiply both sides of the inequality by 2:
```
2(x^2 - 2) < 2(rac{7}{2}x)
2x^2 - 4 < 7x
```
Rearrange the inequality: Now, let's move all the terms to one side to get a standard quadratic form:
```
2x^2 - 7x - 4 < 0
```
Factor the quadratic expression: We need to factor the quadratic expression $2x^2 - 7x - 4$ . Think of two numbers that multiply to $(2)(-4) = -8$ and add up to -7. These numbers are -8 and 1. So, we can rewrite the middle term:
```
2x^2 - 8x + x - 4 < 0
```
Now, factor by grouping:
```
2x(x - 4) + 1(x - 4) < 0
(2x + 1)(x - 4) < 0
```
Find the critical points: The critical points are the values of x that make the expression $(2x + 1)(x - 4)$ equal to zero. These are the points where the quadratic expression changes its sign. Set each factor equal to zero and solve:
```
2x + 1 = 0  =>  x = -rac{1}{2}
x - 4 = 0  =>  x = 4
```
So, our critical points are $x = - rac{1}{2}$ and $x = 4$ .
Test intervals: The critical points divide the number line into three intervals: $(- ext{\infty}, - rac{1}{2})$ , $(- rac{1}{2}, 4)$ , and $(4, ext{∞})$ . We need to test a value from each interval to see where the inequality $(2x + 1)(x - 4) < 0$ holds true.
- Interval $(- ext{\infty}, - rac{1}{2})$ : Let's test $x = -1$ :
```
(2(-1) + 1)(-1 - 4) = (-1)(-5) = 5 > 0
```
  The inequality is not true in this interval.
- Interval $(- rac{1}{2}, 4)$ : Let's test $x = 0$ :
```
(2(0) + 1)(0 - 4) = (1)(-4) = -4 < 0
```
  The inequality is true in this interval.
- Interval $(4, ext{∞})$ : Let's test $x = 5$ :
```
(2(5) + 1)(5 - 4) = (11)(1) = 11 > 0
```
  The inequality is not true in this interval.
Write the solution: The inequality $(2x + 1)(x - 4) < 0$ is true for the interval $(- rac{1}{2}, 4)$ . This means the solution to our original inequality is $- rac{1}{2} < x < 4$ .

Visualizing the Solution

It can be helpful to visualize the solution on a number line. Mark the critical points $- rac{1}{2}$ and 4. Since we have a