Solve The Inequality. Then Graph The Solutions On A Number Line. − 10 \textgreater − 2 ( 3 X − 7 -10 \ \textgreater \ -2(3x - 7 − 10 \textgreater − 2 ( 3 X − 7 ]A. X ≤ □ X \leq \square X ≤ □ B. X ≥ □ X \geq \square X ≥ □ C. X \textless □ X \ \textless \ \square X \textless □ D. X \textgreater □ X \ \textgreater \ \square X \textgreater □

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 isolating the variable on one side of the inequality sign, while graphing inequalities on a number line requires visualizing the solution set. In this article, we will focus on solving and graphing the inequality $-10 \ \textgreater \ -2(3x - 7)$.

Step 1: Distribute the Negative 2

To begin solving the inequality, we need to distribute the negative 2 to the terms inside the parentheses.

$-10 \ \textgreater \ -2(3x - 7)$

$-10 \ \textgreater \ -6x + 14$

Step 2: Add 10 to Both Sides

Next, we add 10 to both sides of the inequality to isolate the term with the variable.

$-10 + 10 \ \textgreater \ -6x + 14 + 10$

$0 \ \textgreater \ -6x + 24$

Step 3: Subtract 24 from Both Sides

Now, we subtract 24 from both sides of the inequality to further isolate the term with the variable.

$0 - 24 \ \textgreater \ -6x + 24 - 24$

$-24 \ \textgreater \ -6x$

Step 4: Divide Both Sides by -6

To solve for x, we need to divide both sides of the inequality by -6. However, when dividing or multiplying an inequality by a negative number, we must reverse the direction of the inequality sign.

$\frac{-24}{-6} \ \textless \ \frac{-6x}{-6}$

$4 \ \textless \ x$

Graphing the Solution on a Number Line

To graph the solution on a number line, we need to visualize the set of all values that satisfy the inequality. Since the inequality is $4 \ \textless \ x$, we draw an open circle at x = 4 and shade the region to the right of the circle.

Conclusion

In conclusion, solving and graphing inequalities require a step-by-step approach. By distributing the negative 2, adding 10 to both sides, subtracting 24 from both sides, and dividing both sides by -6, we can isolate the variable and determine the solution set. Graphing the solution on a number line involves visualizing the set of all values that satisfy the inequality.

Answer Key

A. $x \leq \square$ (Incorrect)

B. $x \geq \square$ (Incorrect)

C. $x \ \textless \ \square$ (Incorrect)

D. $x \ \textgreater \ \square$ (Correct)

Additional Tips and Examples

• When solving inequalities, always check the direction of the inequality sign after dividing or multiplying by a negative number.
• When graphing inequalities on a number line, use open circles for strict inequalities (e.g., $x \ \textless \ 4$) and closed circles for non-strict inequalities (e.g., $x \leq 4$).
• Practice solving and graphing inequalities with different types of expressions, such as linear and quadratic inequalities.

Common Mistakes to Avoid

• Failing to distribute the negative 2 in the original inequality.
• Not reversing the direction of the inequality sign when dividing or multiplying by a negative number.
• Not using open circles for strict inequalities and closed circles for non-strict inequalities when graphing on a number line.

Real-World Applications

• Inequalities are used in various real-world applications, such as finance, economics, and engineering.
• Solving and graphing inequalities can help us make informed decisions and model real-world scenarios.
• Inequalities are used to describe relationships between variables and can be used to optimize functions and make predictions.

Conclusion

Introduction

In our previous article, we explored the step-by-step process of solving and graphing inequalities. In this article, we will address some common questions and concerns that students may have when working with inequalities.

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 $ax + b > c$ or $ax + b < c$, where $a$, $b$, and $c$ are constants. A quadratic inequality, on the other hand, is an inequality that can be written in the form $ax^2 + bx + c > 0$ or $ax^2 + bx + c < 0$, where $a$, $b$, and $c$ are constants.

Q: How do I know whether to use an open circle or a closed circle when graphing an inequality on a number line?

A: When graphing an inequality on a number line, you should use an open circle for strict inequalities (e.g., $x \ \textless \ 4$) and a closed circle for non-strict inequalities (e.g., $x \leq 4$).

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". For example, $x > 2$ and $x < 5$ is a compound inequality. A single inequality, on the other hand, is an inequality that involves only one expression. For example, $x > 3$ is a single inequality.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to solve each inequality separately and then combine the solutions. For example, to solve the compound inequality $x > 2$ and $x < 5$, you would first solve $x > 2$ to get $x > 2$, and then solve $x < 5$ to get $x < 5$. The solution to the compound inequality would be the intersection of the two solutions, which is $2 < x < 5$.

Q: What is the difference between an absolute value inequality and a linear inequality?

A: An absolute value inequality is an inequality that involves the absolute value of a variable. For example, $|x| > 2$ is an absolute value inequality. A linear inequality, on the other hand, is an inequality that can be written in the form $ax + b > c$ or $ax + b < c$, where $a$, $b$, and $c$ are constants.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, 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, to solve the absolute value inequality $|x| > 2$, you would first consider the case where $x > 0$, which would give you $x > 2$. You would then consider the case where $x < 0$, which would give you $x < -2$. The solution to the absolute value inequality would be the union of the two solutions, which is $x < -2$ or $x > 2$.

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 constraints. An inequality, on the other hand, is a mathematical expression that compares two values using greater than (>), less than (<), greater than or equal to (≥), or less than or equal to (≤) symbols.

Q: How do I use inequalities in real-world applications?

A: Inequalities are used in various real-world applications, such as finance, economics, and engineering. For example, you might use inequalities to model the relationship between the cost of a product and its price, or to determine the maximum or minimum value of a function.

Conclusion

In conclusion, inequalities are a fundamental concept in mathematics that have numerous real-world applications. By understanding how to solve and graph inequalities, you can apply this knowledge to a wide range of problems and scenarios. We hope that this Q&A guide has been helpful in addressing some common questions and concerns that students may have when working with inequalities.