Which Represents The Solution Set To The Inequality $-1.5(4x+1) \geq 4.5 - 2.5(x+1$\]?A. $x \geq -1$B. $x \geq \frac{7}{16}$C. $(-\infty, -1\]D. $\left(-\infty, \frac{7}{16}\right\]Solve The Inequality:1.

Introduction

Inequalities are mathematical expressions that compare two values using greater than, less than, greater than or equal to, or less than or equal to. Solving inequalities involves isolating the variable on one side of the inequality sign. In this article, we will focus on solving a specific inequality and provide a step-by-step guide on how to approach it.

The Inequality

The given inequality is:

$$-1.5(4x+1) \geq 4.5 - 2.5(x+1)$$

Our goal is to find the solution set to this inequality, which represents the values of $x$ that satisfy the given inequality.

Step 1: Distribute the Negative 1.5

To start solving the inequality, we need to distribute the negative 1.5 to the terms inside the parentheses.

$$-1.5(4x+1) = -6x - 1.5$$

So, the inequality becomes:

$$-6x - 1.5 \geq 4.5 - 2.5(x+1)$$

Step 2: Simplify the Right-Hand Side

Next, we need to simplify the right-hand side of the inequality by distributing the negative 2.5 to the terms inside the parentheses.

$$4.5 - 2.5(x+1) = 4.5 - 2.5x - 2.5$$

Combining like terms, we get:

$$4.5 - 2.5x - 2.5 = 2 - 2.5x$$

So, the inequality becomes:

$$-6x - 1.5 \geq 2 - 2.5x$$

Step 3: Add 2.5x to Both Sides

To isolate the variable $x$ on one side of the inequality, we need to add 2.5x to both sides of the inequality.

$$-6x + 2.5x - 1.5 \geq 2$$

Combining like terms, we get:

$$-3.5x - 1.5 \geq 2$$

Step 4: Add 1.5 to Both Sides

Next, we need to add 1.5 to both sides of the inequality to get rid of the negative term.

$$-3.5x - 1.5 + 1.5 \geq 2 + 1.5$$

Simplifying, we get:

$$-3.5x \geq 3.5$$

Step 5: Divide Both Sides by -3.5

Finally, we need to divide both sides of the inequality by -3.5 to solve for $x$.

$$\frac{-3.5x}{-3.5} \geq \frac{3.5}{-3.5}$$

Simplifying, we get:

$$x \leq -1$$

Conclusion

The solution set to the inequality $-1.5(4x+1) \geq 4.5 - 2.5(x+1)$ is $x \leq -1$. This means that any value of $x$ less than or equal to -1 satisfies the given inequality.

Answer

The correct answer is:

• A. $x \geq -1$ is incorrect because the solution set is $x \leq -1$.
• B. $x \geq \frac{7}{16}$ is incorrect because the solution set is $x \leq -1$.
• C. $(-\infty, -1]$ is correct because it represents the solution set $x \leq -1$.
• D. $\left(-\infty, \frac{7}{16}\right)$ is incorrect because the solution set is $x \leq -1$.

Final Answer

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Q&A: Solving Inequalities

Q: What is an inequality?

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

Q: How do I solve an inequality?

A: To solve an inequality, you need to isolate the variable on one side of the inequality sign. This involves using inverse operations to get rid of any terms that are being added or subtracted from the variable.

Q: What are some common inverse operations?

A: Some common inverse operations include:

• Multiplying or dividing both sides of the inequality by a number to get rid of a coefficient
• Adding or subtracting the same value to both sides of the inequality to get rid of a constant term
• Using the opposite operation (e.g. addition and subtraction, multiplication and division) to get rid of a term

Q: How do I know which direction to move the inequality sign?

A: When you multiply or divide both sides of the inequality by a negative number, you need to flip the direction of the inequality sign. For example, if you have $x \geq 2$ and you multiply both sides by -3, the inequality becomes $-3x \leq -6$.

Q: What is the solution set to an inequality?

A: The solution set to an inequality is the set of all values of the variable that satisfy the inequality. This can be represented graphically on a number line or as an interval.

Q: How do I graph the solution set to an inequality?

A: To graph the solution set to an inequality, you need to plot a number line and shade in the region that satisfies the inequality. For example, if you have the inequality $x \geq 2$, you would plot a number line and shade in the region to the right of 2.

Q: What are some common types of inequalities?

A: Some common types of inequalities include:

• Linear inequalities: These are inequalities that can be written in the form $ax + b \geq c$ or $ax + b \leq c$.
• Quadratic inequalities: These are inequalities that can be written in the form $ax^2 + bx + c \geq 0$ or $ax^2 + bx + c \leq 0$.
• Absolute value inequalities: These are inequalities that involve absolute value expressions, such as $|x| \geq 2$ or $|x| \leq 2$.

Q: How do I solve absolute value inequalities?

A: To solve absolute value inequalities, you need to consider two cases: one where the expression inside the absolute value is positive, and one where the expression inside the absolute value is negative. For example, if you have the inequality $|x| \geq 2$, you would consider two cases: $x \geq 2$ and $x \leq -2$.

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

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

• Flipping the direction of the inequality sign when multiplying or dividing both sides by a negative number
• Failing to consider all possible cases when solving absolute value inequalities
• Not checking the solution set to make sure it satisfies the original inequality

Conclusion

Solving inequalities can be a challenging task, but with practice and patience, you can become proficient in solving a wide range of inequalities. Remember to always check your solution set to make sure it satisfies the original inequality, and don't be afraid to ask for help if you get stuck.