What Is A Correct First Step In Solving The Inequality − 4 ( 3 − 5 X ) ≥ − 6 X + 9 ? -4(3-5x) \geq -6x + 9? − 4 ( 3 − 5 X ) ≥ − 6 X + 9 ? A. − 12 − 20 X ≤ − 6 X + 9 -12 - 20x \leq -6x + 9 − 12 − 20 X ≤ − 6 X + 9 B. − 12 − 20 X ≥ − 6 X + 9 -12 - 20x \geq -6x + 9 − 12 − 20 X ≥ − 6 X + 9 C. − 12 + 20 X ≤ − 6 X + 9 -12 + 20x \leq -6x + 9 − 12 + 20 X ≤ − 6 X + 9 D. − 12 + 20 X ≥ − 6 X + 9 -12 + 20x \geq -6x + 9 − 12 + 20 X ≥ − 6 X + 9

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Understanding the Basics of Inequalities

Inequalities are mathematical expressions that compare two values, indicating whether one value is greater than, less than, or equal to another value. In this article, we will focus on solving a specific type of inequality, which is a linear inequality. A linear inequality is an inequality that can be written in the form of ax + b ≥ c or ax + b ≤ c, where a, b, and c are constants, and x is the variable.

The Correct First Step in Solving the Inequality

The given inequality is −4(3−5x)≥−6x+9-4(3-5x) \geq -6x + 9. To solve this inequality, we need to follow the correct order of operations. The first step is to distribute the negative 4 to the terms inside the parentheses. This will give us −12+20x≥−6x+9-12 + 20x \geq -6x + 9.

Why is this the Correct First Step?

Distributing the negative 4 to the terms inside the parentheses is the correct first step because it allows us to simplify the inequality and make it easier to solve. By distributing the negative 4, we are essentially multiplying each term inside the parentheses by -4, which will give us a new expression that is equivalent to the original inequality.

Alternative Options

Let's take a closer look at the alternative options:

  • Option A: −12−20x≤−6x+9-12 - 20x \leq -6x + 9: This option is incorrect because it changes the direction of the inequality, which is not allowed when distributing a negative number.
  • Option B: −12−20x≥−6x+9-12 - 20x \geq -6x + 9: This option is also incorrect for the same reason as Option A.
  • Option D: −12+20x≥−6x+9-12 + 20x \geq -6x + 9: This option is correct, but it is not the simplest form of the inequality. We can simplify it further by combining like terms.

Simplifying the Inequality

Now that we have distributed the negative 4, we can simplify the inequality by combining like terms. To do this, we need to move all the terms with x to one side of the inequality and all the constant terms to the other side.

Step 1: Move all the terms with x to one side

We can start by subtracting -6x from both sides of the inequality. This will give us 20x+6x≥920x + 6x \geq 9. Simplifying further, we get 26x≥926x \geq 9.

Step 2: Move all the constant terms to the other side

Next, we can subtract 9 from both sides of the inequality. This will give us 26x≥9−926x \geq 9 - 9, which simplifies to 26x≥026x \geq 0.

Step 3: Divide both sides by 26

Finally, we can divide both sides of the inequality by 26. This will give us x≥0/26x \geq 0/26, which simplifies to x≥0x \geq 0.

Conclusion

In conclusion, the correct first step in solving the inequality −4(3−5x)≥−6x+9-4(3-5x) \geq -6x + 9 is to distribute the negative 4 to the terms inside the parentheses. This will give us −12+20x≥−6x+9-12 + 20x \geq -6x + 9. From there, we can simplify the inequality by combining like terms and dividing both sides by 26. The final solution is x≥0x \geq 0.

Key Takeaways

  • The correct first step in solving a linear inequality is to distribute any negative numbers to the terms inside the parentheses.
  • When distributing a negative number, the direction of the inequality may change.
  • To simplify an inequality, we can combine like terms and divide both sides by a non-zero constant.
  • The final solution to an inequality should be in the simplest form possible.

Practice Problems

  1. Solve the inequality 2(x+3)≥5x−22(x + 3) \geq 5x - 2.
  2. Solve the inequality −3(2x−1)≤x+4-3(2x - 1) \leq x + 4.
  3. Solve the inequality x/2+1≥3x−2x/2 + 1 \geq 3x - 2.

Answer Key

  1. x≥5x \geq 5
  2. x≤2x \leq 2
  3. x≥4x \geq 4

Additional Resources

For more practice problems and additional resources, check out the following websites:

  • Khan Academy: Inequalities
  • Mathway: Inequality Solver
  • Wolfram Alpha: Inequality Solver

By following the steps outlined in this article, you should be able to solve linear inequalities with ease. Remember to always distribute negative numbers to the terms inside the parentheses and simplify the inequality by combining like terms and dividing both sides by a non-zero constant. Happy solving!

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Q: What is the first step in solving a linear inequality?

A: The first step in solving a linear inequality is to distribute any negative numbers to the terms inside the parentheses. This will give you a new expression that is equivalent to the original inequality.

Q: How do I simplify an inequality?

A: To simplify an inequality, you can combine like terms and divide both sides by a non-zero constant. This will give you a new expression that is equivalent to the original inequality, but in a simpler form.

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 of ax + b ≥ c or ax + b ≤ c, where a, b, and c are constants, and x is the variable. A quadratic inequality, on the other hand, is an inequality that can be written in the form of ax^2 + bx + c ≥ d or ax^2 + bx + c ≤ d, where a, b, c, and d are constants, and x is the variable.

Q: How do I solve a quadratic inequality?

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

  1. Factor the quadratic expression, if possible.
  2. Set each factor equal to zero and solve for x.
  3. Use a number line or a graph to determine the intervals where the inequality is true.

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

A: A strict inequality is an inequality that uses the symbols < or >, such as x < 2 or x > 3. A non-strict inequality, on the other hand, uses the symbols ≤ or ≥, such as x ≤ 2 or x ≥ 3.

Q: How do I determine the direction of the inequality?

A: To determine the direction of the inequality, you can use the following steps:

  1. Check if the coefficient of the x term is positive or negative.
  2. If the coefficient is positive, the inequality is non-strict (≤ or ≥).
  3. If the coefficient is negative, the inequality is strict (< or >).

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

A: A linear inequality is an inequality that can be written in the form of ax + b ≥ c or ax + b ≤ c, where a, b, and c are constants, and x is the variable. A rational inequality, on the other hand, is an inequality that can be written in the form of a(x)/b(x) ≥ c or a(x)/b(x) ≤ c, where a(x), b(x), and c are polynomials, and x is the variable.

Q: How do I solve a rational inequality?

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

  1. Factor the numerator and denominator, if possible.
  2. Cancel out any common factors.
  3. Set the numerator or denominator equal to zero and solve for x.
  4. Use a number line or a graph to determine the intervals where the inequality is true.

Q: What are some common mistakes to avoid when solving inequalities?

A: Some common mistakes to avoid when solving inequalities include:

  • Forgetting to distribute negative numbers to the terms inside the parentheses.
  • Not combining like terms.
  • Not dividing both sides by a non-zero constant.
  • Not checking the direction of the inequality.
  • Not using a number line or graph to determine the intervals where the inequality is true.

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through practice problems, such as those found in a textbook or online resource. You can also try solving inequalities on your own, using a calculator or computer program to check your work.

Q: What are some real-world applications of inequalities?

A: Inequalities have many real-world applications, including:

  • Finance: Inequalities are used to calculate interest rates and investment returns.
  • Science: Inequalities are used to model population growth and decay.
  • Engineering: Inequalities are used to design and optimize systems.
  • Economics: Inequalities are used to model supply and demand.

By following the steps outlined in this article, you should be able to solve inequalities with ease. Remember to always distribute negative numbers to the terms inside the parentheses, combine like terms, and divide both sides by a non-zero constant. Happy solving!