Solve The Inequality:$\[ \frac{d+5}{-2} \leq -\frac{3}{2} \\]

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Introduction

Inequalities are mathematical expressions that compare two values, often using greater than, less than, greater than or equal to, or less than or equal to symbols. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving the inequality d+5−2≤−32\frac{d+5}{-2} \leq -\frac{3}{2}.

Understanding the Inequality

The given inequality is d+5−2≤−32\frac{d+5}{-2} \leq -\frac{3}{2}. To solve this inequality, we need to isolate the variable dd on one side of the inequality. We can start by multiplying both sides of the inequality by −2-2 to eliminate the fraction.

Multiplying Both Sides by -2

When we multiply both sides of the inequality by −2-2, we need to remember that this will change the direction of the inequality. Since we are multiplying by a negative number, the inequality will become ≥\geq instead of ≤\leq.

\frac{d+5}{-2} \leq -\frac{3}{2}
\\
\Rightarrow (d+5) \geq 3

Isolating the Variable

Now that we have eliminated the fraction, we can isolate the variable dd by subtracting 55 from both sides of the inequality.

Subtracting 5 from Both Sides

(d+5) \geq 3
\\
\Rightarrow d \geq -2

Conclusion

In conclusion, the solution to the inequality d+5−2≤−32\frac{d+5}{-2} \leq -\frac{3}{2} is d≥−2d \geq -2. This means that any value of dd that is greater than or equal to −2-2 will make the inequality true.

Tips and Tricks

When solving inequalities, it's essential to remember the following tips and tricks:

  • Direction of Inequality: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality will change.
  • Isolating the Variable: To solve an inequality, we need to isolate the variable on one side of the inequality.
  • Checking the Solution: After solving an inequality, it's essential to check the solution by plugging in a value of the variable that satisfies the inequality.

Real-World Applications

Solving inequalities has numerous real-world applications in various fields, including:

  • Science: Inequalities are used to model real-world phenomena, such as population growth, chemical reactions, and physical systems.
  • Engineering: Inequalities are used to design and optimize systems, such as electrical circuits, mechanical systems, and computer networks.
  • Economics: Inequalities are used to model economic systems, such as supply and demand, inflation, and unemployment.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid the following common mistakes:

  • Incorrect Direction of Inequality: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality will change.
  • Not Isolating the Variable: To solve an inequality, we need to isolate the variable on one side of the inequality.
  • Not Checking the Solution: After solving an inequality, it's essential to check the solution by plugging in a value of the variable that satisfies the inequality.

Conclusion

In conclusion, solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the tips and tricks outlined in this article, you can master the art of solving inequalities and apply it to various fields. Remember to avoid common mistakes and always check your solution to ensure that it's correct.

Frequently Asked Questions

Q: What is an inequality?

A: An inequality is a mathematical expression that compares two values, often using greater than, less than, greater than or equal to, or less than or equal to symbols.

Q: How do I solve an inequality?

A: To solve an inequality, we need to isolate the variable on one side of the inequality. We can do this by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

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

A: A linear inequality is an inequality that can be written in the form ax+b≤cax + b \leq c, where aa, bb, and cc are constants. A quadratic inequality is an inequality that can be written in the form ax2+bx+c≤0ax^2 + bx + c \leq 0, where aa, bb, and cc are constants.

Q: How do I check the solution to an inequality?

A: To check the solution to an inequality, we need to plug in a value of the variable that satisfies the inequality and verify that the inequality is true.

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Q&A: Solving Inequalities

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to isolate the variable on one side of the inequality. This can be done by adding or subtracting the same value from both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: How do I handle fractions in an inequality?

A: When an inequality contains fractions, we can eliminate the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators. This will allow us to isolate the variable and solve the inequality.

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

A: A linear inequality is an inequality that can be written in the form ax+b≤cax + b \leq c, where aa, bb, and cc are constants. A quadratic inequality is an inequality that can be written in the form ax2+bx+c≤0ax^2 + bx + c \leq 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, we need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution set. We can also use the quadratic formula to find the solutions to the quadratic equation.

Q: What is the importance of checking the solution to an inequality?

A: Checking the solution to an inequality is essential to ensure that the solution is correct. By plugging in a value of the variable that satisfies the inequality, we can verify that the inequality is true and that the solution is correct.

Q: How do I graph an inequality on a number line?

A: To graph an inequality on a number line, we need to plot the values that satisfy the inequality. We can use a closed circle to represent the values that satisfy the inequality, and a open circle to represent the values that do not satisfy the inequality.

Q: What is the difference between a compound inequality and a single inequality?

A: A compound inequality is an inequality that contains two or more inequalities joined by the word "and" or "or". A single inequality is an inequality that contains only one inequality.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, we need to solve each inequality separately and then combine the solutions. We can use the word "and" to combine the solutions, or the word "or" to combine the solutions.

Q: What is the importance of understanding the concept of absolute value in inequalities?

A: Understanding the concept of absolute value is essential in solving inequalities. Absolute value inequalities involve the absolute value of a variable, and can be solved using the properties of absolute value.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, we need to isolate the variable and then use the properties of absolute value to determine the solution set. We can also use the definition of absolute value to solve the inequality.

Q: What is the difference between a linear programming problem and an inequality?

A: A linear programming problem is a problem that involves maximizing or minimizing a linear function subject to a set of linear inequalities. An inequality is a mathematical expression that compares two values, often using greater than, less than, greater than or equal to, or less than or equal to symbols.

Q: How do I solve a linear programming problem?

A: To solve a linear programming problem, we need to use the method of linear programming to find the optimal solution. This involves setting up a system of linear inequalities and then using the simplex method or other methods to find the optimal solution.

Conclusion

In conclusion, solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the tips and tricks outlined in this article, you can master the art of solving inequalities and apply it to various fields. Remember to avoid common mistakes and always check your solution to ensure that it's correct.

Frequently Asked Questions

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to isolate the variable on one side of the inequality.

Q: How do I handle fractions in an inequality?

A: When an inequality contains fractions, we can eliminate the fractions by multiplying both sides of the inequality by the least common multiple (LCM) of the denominators.

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

A: A linear inequality is an inequality that can be written in the form ax+b≤cax + b \leq c, where aa, bb, and cc are constants. A quadratic inequality is an inequality that can be written in the form ax2+bx+c≤0ax^2 + bx + c \leq 0, where aa, bb, and cc are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, we need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution set.

Real-World Applications

Solving inequalities has numerous real-world applications in various fields, including:

  • Science: Inequalities are used to model real-world phenomena, such as population growth, chemical reactions, and physical systems.
  • Engineering: Inequalities are used to design and optimize systems, such as electrical circuits, mechanical systems, and computer networks.
  • Economics: Inequalities are used to model economic systems, such as supply and demand, inflation, and unemployment.

Common Mistakes to Avoid

When solving inequalities, it's essential to avoid the following common mistakes:

  • Incorrect Direction of Inequality: When multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality will change.
  • Not Isolating the Variable: To solve an inequality, we need to isolate the variable on one side of the inequality.
  • Not Checking the Solution: After solving an inequality, it's essential to check the solution by plugging in a value of the variable that satisfies the inequality.

Conclusion

In conclusion, solving inequalities is a crucial skill in mathematics that has numerous real-world applications. By following the tips and tricks outlined in this article, you can master the art of solving inequalities and apply it to various fields. Remember to avoid common mistakes and always check your solution to ensure that it's correct.