Solve The Following Inequalities:a) $x^2 - 5x \ \textless \ 6$b) $4x^2 \ \textgreater \ 9$c) $2x^2 - 5x - 3 \ \textless \ 0$d) $x^2 - X \ \textless \ 12$e) $(1 - 2x)(x + 3) \leqslant 0$f) $3x + 9

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In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more mathematical expressions. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will explore the solution to various types of inequalities, including quadratic inequalities, linear inequalities, and rational inequalities.

Quadratic Inequalities

Quadratic inequalities are inequalities that involve 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.

Solving Quadratic Inequalities

To solve a quadratic inequality, we can use the following steps:

  1. Factor the quadratic expression: If the quadratic expression can be factored, we can write it as a product of two binomials.
  2. Find the roots of the quadratic expression: The roots of the quadratic expression are the values of x that make the expression equal to zero.
  3. Use the roots to determine the sign of the quadratic expression: If the quadratic expression is positive, it is greater than zero. If it is negative, it is less than zero.
  4. Determine the solution set: The solution set is the set of all values of x that satisfy the inequality.

Example 1: Solving the Inequality x^2 - 5x < 6

To solve this inequality, we can start by factoring the quadratic expression:

x^2 - 5x - 6 = (x - 6)(x + 1)

Next, we can find the roots of the quadratic expression:

x - 6 = 0 --> x = 6

x + 1 = 0 --> x = -1

Now, we can use the roots to determine the sign of the quadratic expression:

For x < -1, both (x - 6) and (x + 1) are negative, so the product is positive.

For -1 < x < 6, (x - 6) is negative and (x + 1) is positive, so the product is negative.

For x > 6, both (x - 6) and (x + 1) are positive, so the product is positive.

Therefore, the solution set is:

x < -1 or x > 6

Example 2: Solving the Inequality 4x^2 > 9

To solve this inequality, we can start by dividing both sides by 4:

x^2 > 9/4

Next, we can take the square root of both sides:

x > ±√(9/4)

x > ±3/2

Therefore, the solution set is:

x > 3/2 or x < -3/2

Linear Inequalities

Linear inequalities are inequalities that involve a linear expression. The general form of a linear inequality is:

ax + b < c

or

ax + b > c

where a, b, and c are constants.

Solving Linear Inequalities

To solve a linear inequality, we can use the following steps:

  1. Isolate the variable: We can isolate the variable x by subtracting b from both sides and then dividing both sides by a.
  2. Determine the solution set: The solution set is the set of all values of x that satisfy the inequality.

Example 3: Solving the Inequality 3x + 9 > 0

To solve this inequality, we can start by isolating the variable x:

3x > -9

Next, we can divide both sides by 3:

x > -3

Therefore, the solution set is:

x > -3

Rational Inequalities

Rational inequalities are inequalities that involve a rational expression. The general form of a rational inequality is:

f(x)/g(x) < 0

or

f(x)/g(x) > 0

where f(x) and g(x) are rational expressions.

Solving Rational Inequalities

To solve a rational inequality, we can use the following steps:

  1. Find the zeros of the numerator and denominator: The zeros of the numerator and denominator are the values of x that make the expression equal to zero.
  2. Use the zeros to determine the sign of the rational expression: If the rational expression is positive, it is greater than zero. If it is negative, it is less than zero.
  3. Determine the solution set: The solution set is the set of all values of x that satisfy the inequality.

Example 4: Solving the Inequality (1 - 2x)(x + 3) ≤ 0

To solve this inequality, we can start by finding the zeros of the numerator and denominator:

1 - 2x = 0 --> x = 1/2

x + 3 = 0 --> x = -3

Next, we can use the zeros to determine the sign of the rational expression:

For x < -3, both (1 - 2x) and (x + 3) are negative, so the product is positive.

For -3 < x < 1/2, (1 - 2x) is negative and (x + 3) is positive, so the product is negative.

For x > 1/2, both (1 - 2x) and (x + 3) are positive, so the product is positive.

Therefore, the solution set is:

x ≤ -3 or -3 < x ≤ 1/2

Conclusion

In this article, we have explored the solution to various types of inequalities, including quadratic inequalities, linear inequalities, and rational inequalities. We have used the following steps to solve each type of inequality:

  • Factor the quadratic expression
  • Find the roots of the quadratic expression
  • Use the roots to determine the sign of the quadratic expression
  • Determine the solution set

We have also used the following steps to solve linear inequalities:

  • Isolate the variable
  • Determine the solution set

Finally, we have used the following steps to solve rational inequalities:

  • Find the zeros of the numerator and denominator
  • Use the zeros to determine the sign of the rational expression
  • Determine the solution set

In the previous article, we explored the solution to various types of inequalities, including quadratic inequalities, linear inequalities, and rational inequalities. In this article, we will answer some frequently asked questions about solving inequalities.

Q: What is the difference between a quadratic inequality and a linear inequality?

A: A quadratic inequality is an inequality that involves a quadratic expression, while a linear inequality is an inequality that involves a linear expression. For example, x^2 - 5x < 6 is a quadratic inequality, while 3x + 9 > 0 is a linear inequality.

Q: How do I determine the sign of a quadratic expression?

A: To determine the sign of a quadratic expression, you can use the following steps:

  1. Factor the quadratic expression: If the quadratic expression can be factored, you can write it as a product of two binomials.
  2. Find the roots of the quadratic expression: The roots of the quadratic expression are the values of x that make the expression equal to zero.
  3. Use the roots to determine the sign of the quadratic expression: If the quadratic expression is positive, it is greater than zero. If it is negative, it is less than zero.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you can use the following steps:

  1. Find the zeros of the numerator and denominator: The zeros of the numerator and denominator are the values of x that make the expression equal to zero.
  2. Use the zeros to determine the sign of the rational expression: If the rational expression is positive, it is greater than zero. If it is negative, it is less than zero.
  3. Determine the solution set: The solution set is the set of all values of x that satisfy the inequality.

Q: What is the difference between a strict inequality and a non-strict inequality?

A: A strict inequality is an inequality that is written with a strict symbol, such as < or >. A non-strict inequality is an inequality that is written with a non-strict symbol, such as ≤ or ≥. For example, x^2 - 5x < 6 is a strict inequality, while x^2 - 5x ≤ 6 is a non-strict inequality.

Q: How do I determine the solution set of an inequality?

A: To determine the solution set of an inequality, you can use the following steps:

  1. Determine the sign of the expression: If the expression is positive, it is greater than zero. If it is negative, it is less than zero.
  2. Determine the intervals where the expression is positive or negative: The intervals where the expression is positive or negative are the solution set.

Q: Can I use a graph to solve an inequality?

A: Yes, you can use a graph to solve an inequality. A graph can help you visualize the solution set and determine the intervals where the expression is positive or negative.

Q: What is the importance of solving inequalities?

A: Solving inequalities is an important skill in mathematics because it allows you to determine the solution set of an inequality. The solution set is the set of all values of x that satisfy the inequality. Solving inequalities is also important in real-world applications, such as optimization problems and data analysis.

Q: Can I use a calculator to solve an inequality?

A: Yes, you can use a calculator to solve an inequality. Many calculators have built-in functions that can help you solve inequalities, such as the quadratic formula and the rational root theorem.

Conclusion

In this article, we have answered some frequently asked questions about solving inequalities. We have covered topics such as the difference between a quadratic inequality and a linear inequality, how to determine the sign of a quadratic expression, and how to solve a rational inequality. We have also discussed the importance of solving inequalities and how to use a graph to solve an inequality. By following these steps and using the right tools, you can solve a wide range of inequalities and determine the solution set.

Additional Resources

  • Quadratic Formula: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a
  • Rational Root Theorem: The rational root theorem is a mathematical theorem that can be used to find the rational roots of a polynomial equation. It states that if a polynomial equation has a rational root, then that root must be a factor of the constant term.
  • Graphing Calculator: A graphing calculator is a type of calculator that can be used to graph functions and solve equations. It is a powerful tool that can be used to visualize the solution set of an inequality.

By using these resources and following the steps outlined in this article, you can solve a wide range of inequalities and determine the solution set.