Solve The Inequality:$\[ |6 - 2x| - 6 \ \textgreater \ 4 \\]

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Introduction

Inequalities are mathematical expressions that compare two values, often with a greater-than or less-than symbol. Solving inequalities involves finding the values of the variable that make the inequality true. In this article, we will focus on solving the inequality $|6 - 2x| - 6 > 4$. We will break down the solution into manageable steps, using algebraic manipulations and logical reasoning.

Understanding Absolute Value

Before we dive into solving the inequality, let's review the concept of absolute value. The absolute value of a number $a$, denoted by $|a|$, is the distance of $a$ from zero on the number line. In other words, it is the magnitude of $a$ without considering its direction. For example, $|5| = 5$ and $|-5| = 5$. Absolute value is often used to represent the distance between two points on the number line.

Step 1: Isolate the Absolute Value Expression

To solve the inequality $|6 - 2x| - 6 > 4$, we need to isolate the absolute value expression. We can do this by adding 6 to both sides of the inequality:

$$|6 - 2x| > 10$$

Step 2: Split the Inequality into Two Cases

When dealing with absolute value inequalities, we need to consider two cases: one where the expression inside the absolute value is positive, and another where it is negative. Let's split the inequality into two cases:

Case 1: $6 - 2x > 0$

In this case, the absolute value expression is equal to $6 - 2x$. We can rewrite the inequality as:

$$6 - 2x > 10$$

Subtracting 6 from both sides gives us:

$$-2x > 4$$

Dividing both sides by -2 (and flipping the inequality sign) gives us:

$$x < -2$$

Case 2: $6 - 2x < 0$

In this case, the absolute value expression is equal to $-(6 - 2x)$. We can rewrite the inequality as:

$$-(6 - 2x) > 10$$

Distributing the negative sign gives us:

$$-6 + 2x > 10$$

Adding 6 to both sides gives us:

$$2x > 16$$

Dividing both sides by 2 gives us:

$$x > 8$$

Combining the Cases

We have two cases: $x < -2$ and $x > 8$. To find the solution to the original inequality, we need to combine these cases. Since the two cases are mutually exclusive (i.e., they cannot occur at the same time), we can simply list the two cases as separate solutions:

Solution: $x < -2$ or $x > 8$

Conclusion

Solving the inequality $|6 - 2x| - 6 > 4$ involves isolating the absolute value expression, splitting the inequality into two cases, and combining the cases. By following these steps, we can find the solution to the inequality, which is $x < -2$ or $x > 8$. This solution represents the set of all values of $x$ that make the inequality true.

Example Applications

Inequalities are used in a wide range of applications, including:

• Optimization problems: Inequalities are used to find the maximum or minimum value of a function subject to certain constraints.
• Data analysis: Inequalities are used to analyze and interpret data, such as finding the range of values for a particular variable.
• Engineering: Inequalities are used to design and optimize systems, such as electrical circuits and mechanical systems.

Practice Problems

To practice solving inequalities, try the following problems:

• Solve the inequality $|3x - 2| > 5$.
• Solve the inequality $|2x + 1| < 3$.
• Solve the inequality $|x - 4| > 2$.

Glossary

• Absolute value: The distance of a number from zero on the number line.
• Inequality: A mathematical expression that compares two values, often with a greater-than or less-than symbol.
• Solution: The set of all values of the variable that make the inequality true.

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

A: The first step in solving an inequality involving absolute value is to isolate the absolute value expression. This involves moving any constants or variables that are not part of the absolute value expression to the other side of the inequality.

Q: How do I split an inequality into two cases?

A: To split an inequality into two cases, you need to consider two scenarios: one where the expression inside the absolute value is positive, and another where it is negative. You can use the following steps to split the inequality:

1. Determine the critical point(s) of the inequality, which is the value(s) that makes the expression inside the absolute value equal to zero.
2. Split the inequality into two cases: one where the expression inside the absolute value is positive, and another where it is negative.
3. Solve each case separately, using the same steps as before.

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

A: A linear inequality is an inequality that involves a linear expression, such as $x > 2$ or $x < -3$. A quadratic inequality, on the other hand, involves a quadratic expression, such as $x^2 + 4x + 4 > 0$ or $x^2 - 6x + 9 < 0$.

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 the variable.
3. Use the solutions to create intervals on the number line.
4. Test each interval to determine which ones satisfy the inequality.

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, such as $x > 2$ and $x < 5$. A single inequality, on the other hand, involves a single inequality, such as $x > 3$.

Q: How do I solve a compound inequality?

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

1. Solve each inequality separately, using the same steps as before.
2. Combine the solutions to create a single solution set.

Q: What is the difference between an open interval and a closed interval?

A: An open interval is an interval that does not include the endpoints, such as $(2, 5)$. A closed interval, on the other hand, includes the endpoints, such as $[2, 5]$.

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

A: To graph an inequality on a number line, you can use the following steps:

1. Determine the critical point(s) of the inequality.
2. Plot the critical point(s) on the number line.
3. Test each interval to determine which ones satisfy the inequality.
4. Shade the intervals that satisfy the inequality.

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

A: A linear programming problem is a problem that involves maximizing or minimizing a linear function subject to linear constraints. A quadratic programming problem, on the other hand, involves maximizing or minimizing a quadratic function subject to quadratic constraints.

Q: How do I solve a linear programming problem?

A: To solve a linear programming problem, you can use the following steps:

1. Define the objective function and the constraints.
2. Graph the constraints on a coordinate plane.
3. Find the feasible region, which is the region that satisfies all the constraints.
4. Find the optimal solution, which is the point in the feasible region that maximizes or minimizes the objective function.

Q: What is the difference between a system of linear inequalities and a system of quadratic inequalities?

A: A system of linear inequalities is a set of linear inequalities that involve the same variable(s). A system of quadratic inequalities, on the other hand, involves quadratic inequalities that involve the same variable(s).

Q: How do I solve a system of linear inequalities?

A: To solve a system of linear inequalities, you can use the following steps:

1. Graph each inequality on a coordinate plane.
2. Find the intersection of the graphs, which is the region that satisfies all the inequalities.
3. Test each point in the intersection to determine which ones satisfy the system.

Q: What is the difference between a system of quadratic inequalities and a system of rational inequalities?

A: A system of quadratic inequalities is a set of quadratic inequalities that involve the same variable(s). A system of rational inequalities, on the other hand, involves rational inequalities that involve the same variable(s).

Q: How do I solve a system of rational inequalities?

A: To solve a system of rational inequalities, you can use the following steps:

1. Graph each inequality on a coordinate plane.
2. Find the intersection of the graphs, which is the region that satisfies all the inequalities.
3. Test each point in the intersection to determine which ones satisfy the system.

By following these steps and practicing with different types of inequalities, you can become proficient in solving inequalities and applying them to real-world problems.