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 × 5 − 6 X + 9 -12 + 20 \times 5 - 6x + 9 − 12 + 20 × 5 − 6 X + 9 D. − 12 + 20 X ≥ − 6 X + 9 -12 + 20x \geq -6x + 9 − 12 + 20 X ≥ − 6 X + 9

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 involves simplifying and isolating the variable. Our goal is to find the correct first step in solving the inequality $-4(3-5x) \geq -6x + 9$.

The Importance of Following the Order of Operations

When solving inequalities, it is essential to follow the order of operations (PEMDAS). This means that we need to evaluate expressions inside parentheses first, followed by exponents, multiplication and division, and finally addition and subtraction. In the given inequality, we have an expression inside parentheses that needs to be evaluated first.

Evaluating the Expression Inside Parentheses

Let's start by evaluating the expression inside the parentheses: $3-5x$. This expression can be simplified by distributing the negative sign to the terms inside the parentheses: $-5x - 3$.

Rewriting the Inequality

Now that we have evaluated the expression inside the parentheses, we can rewrite the inequality as follows:

$$-4(-5x - 3) \geq -6x + 9$$

Distributing the Negative Sign

When we distribute the negative sign to the terms inside the parentheses, we need to change the sign of each term. In this case, we have:

$$4(5x + 3) \geq -6x + 9$$

Simplifying the Expression

Now that we have distributed the negative sign, we can simplify the expression by multiplying the terms inside the parentheses:

$$20x + 12 \geq -6x + 9$$

Isolating the Variable

Our goal is to isolate the variable $x$ on one side of the inequality. To do this, we need to get all the terms with $x$ on one side of the inequality. We can do this by adding $6x$ to both sides of the inequality:

$$26x + 12 \geq 9$$

Subtracting 12 from Both Sides

Now that we have isolated the variable $x$ on one side of the inequality, we can subtract 12 from both sides of the inequality to get:

$$26x \geq -3$$

Dividing Both Sides by 26

Finally, we can divide both sides of the inequality by 26 to get:

$$x \geq -\frac{3}{26}$$

Conclusion

In conclusion, the correct first step in solving the inequality $-4(3-5x) \geq -6x + 9$ is to evaluate the expression inside the parentheses and then distribute the negative sign. By following these steps, we can simplify the expression and isolate the variable $x$ on one side of the inequality. The final solution is $x \geq -\frac{3}{26}$.

Answer

Q&A: Solving Inequalities

Q: What is the first step in solving an inequality? A: The first step in solving an inequality is to evaluate the expression inside any parentheses and then distribute the negative sign.

Q: Why is it essential to follow the order of operations when solving inequalities? A: Following the order of operations (PEMDAS) is crucial when solving inequalities because it ensures that we evaluate expressions inside parentheses first, followed by exponents, multiplication and division, and finally addition and subtraction.

Q: How do I distribute the negative sign in an inequality? A: When distributing the negative sign in an inequality, we need to change the sign of each term inside the parentheses. For example, if we have $-4(3-5x)$, we would distribute the negative sign to get $-4(-5x - 3)$.

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 $ax + b \geq c$ or $ax + b \leq c$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c \geq 0$ or $ax^2 + bx + c \leq 0$, where $a$, $b$, and $c$ 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 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 difference between a compound inequality and a single inequality? A: A compound inequality is an inequality that involves two or more inequalities joined by the word "and" or "or". A single inequality, on the other hand, is an inequality that involves only one expression.

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 using the word "and" or "or".

Q: What is the importance of checking the solution set in an inequality? A: Checking the solution set in an inequality is crucial because it ensures that the solution set is correct and that it satisfies the original inequality.

Q: How do I check the solution set in an inequality? A: To check the solution set in an inequality, we need to substitute the solution into the original inequality and verify that it is true.

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

• Not following the order of operations
• Not distributing the negative sign correctly
• Not factoring the quadratic expression correctly
• Not checking the solution set

Conclusion

In conclusion, solving inequalities requires a step-by-step approach that involves evaluating expressions inside parentheses, distributing the negative sign, and isolating the variable. By following these steps and avoiding common mistakes, we can solve inequalities with confidence and accuracy.

Additional Resources

• Khan Academy: Solving Inequalities
• Mathway: Solving Inequalities
• Wolfram Alpha: Solving Inequalities

Practice Problems

1. Solve the inequality $2x + 5 \geq 3x - 2$
2. Solve the inequality $x^2 + 4x + 4 \leq 0$
3. Solve the compound inequality $2x - 3 > 0$ and $x + 2 < 0$

Answer Key

1. $x \geq -3$
2. $x \leq -2$
3. $x < -2$