Let's Solve The Inequality $-x^2 + 7x - 12 \ \textgreater \ 0$.1. For Which Values Of The Variable $x$ Is The Expression $-x^2 + 7x - 12$ Positive?2. For Which Values Of The Variable $x$ Is The Expression $-x^2

Solving the Inequality $-x^2 + 7x - 12 > 0$

In this article, we will delve into the world of quadratic inequalities and solve the inequality $-x^2 + 7x - 12 > 0$. We will explore the values of the variable $x$ for which the expression $-x^2 + 7x - 12$ is positive. This involves factoring the quadratic expression, identifying the critical points, and analyzing the sign of the expression in different intervals.

Understanding Quadratic Inequalities

A quadratic inequality is an inequality that involves a quadratic expression. The general form of a quadratic inequality is $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants. In this case, we have the inequality $-x^2 + 7x - 12 > 0$.

Factoring the Quadratic Expression

To solve the inequality, we need to factor the quadratic expression $-x^2 + 7x - 12$. We can start by finding two numbers whose product is $-12$ and whose sum is $7$. These numbers are $8$ and $-1.5$. Therefore, we can write the quadratic expression as $-(x^2 - 8x + 1.5x - 12)$.

Simplifying the Expression

We can simplify the expression by combining like terms. This gives us $-(x^2 - 8x + 1.5x - 12) = -(x(x - 8) + 1.5(x - 8)) = -(x - 8)(x + 1.5)$.

Identifying Critical Points

The critical points of the inequality are the values of $x$ that make the expression $-(x - 8)(x + 1.5)$ equal to zero. These points are $x = 8$ and $x = -1.5$.

Analyzing the Sign of the Expression

To determine the sign of the expression $-(x - 8)(x + 1.5)$, we can use a sign chart. We will examine the sign of the expression in different intervals of $x$.

Interval 1: $x < -1.5$

In this interval, both $(x - 8)$ and $(x + 1.5)$ are negative. Therefore, the expression $-(x - 8)(x + 1.5)$ is positive.

Interval 2: $-1.5 < x < 8$

In this interval, $(x - 8)$ is negative and $(x + 1.5)$ is positive. Therefore, the expression $-(x - 8)(x + 1.5)$ is negative.

Interval 3: $x > 8$

In this interval, both $(x - 8)$ and $(x + 1.5)$ are positive. Therefore, the expression $-(x - 8)(x + 1.5)$ is negative.

In conclusion, the expression $-x^2 + 7x - 12$ is positive when $x < -1.5$ or $x > 8$. This means that the inequality $-x^2 + 7x - 12 > 0$ is satisfied when $x$ is in the intervals $(-\infty, -1.5)$ or $(8, \infty)$.

The final answer is $\boxed{(-\infty, -1.5) \cup (8, \infty)}$.
Solving the Inequality $-x^2 + 7x - 12 > 0$: Q&A

In our previous article, we solved the inequality $-x^2 + 7x - 12 > 0$ and found that the expression $-x^2 + 7x - 12$ is positive when $x < -1.5$ or $x > 8$. In this article, we will answer some frequently asked questions about solving quadratic inequalities.

Q: What is a quadratic inequality?

A quadratic inequality is an inequality that involves a quadratic expression. The general form of a quadratic inequality is $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants.

Q: How do I factor a quadratic expression?

To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, to factor the quadratic expression $x^2 + 5x + 6$, you need to find two numbers whose product is $6$ and whose sum is $5$. These numbers are $2$ and $3$, so you can write the quadratic expression as $(x + 2)(x + 3)$.

Q: What are critical points?

Critical points are the values of $x$ that make the quadratic expression equal to zero. These points are important because they divide the number line into intervals where the quadratic expression is either positive or negative.

Q: How do I use a sign chart to analyze the sign of a quadratic expression?

A sign chart is a table that shows the sign of the quadratic expression in different intervals of $x$. To use a sign chart, you need to identify the critical points and then examine the sign of the quadratic expression in each interval.

Example: Sign Chart for $-(x - 8)(x + 1.5)$

Q: How do I determine the solution to a quadratic inequality?

To determine the solution to a quadratic inequality, you need to identify the critical points and then examine the sign of the quadratic expression in each interval. The solution is the set of all values of $x$ for which the quadratic expression is positive.

Q: What is the final answer to the inequality $-x^2 + 7x - 12 > 0$?

The final answer to the inequality $-x^2 + 7x - 12 > 0$ is $\boxed{(-\infty, -1.5) \cup (8, \infty)}$.

In conclusion, solving quadratic inequalities involves factoring the quadratic expression, identifying critical points, and analyzing the sign of the expression in different intervals. By using a sign chart and examining the sign of the expression in each interval, you can determine the solution to a quadratic inequality.

• Always factor the quadratic expression before solving the inequality.
• Identify the critical points and examine the sign of the expression in each interval.
• Use a sign chart to analyze the sign of the expression in different intervals.
• The solution to a quadratic inequality is the set of all values of $x$ for which the quadratic expression is positive.