Which Of The Following Values Are Solutions To The Inequality − 8 \textless 2 X + 2 -8 \ \textless \ 2x + 2 − 8 \textless 2 X + 2 ?I. -5 II. -10 III. -4 A. None B. I Only C. II Only D. III Only E. I And II F. I And III G. II And III H. I, II, And III

Understanding the Inequality

The given inequality is $-8 \ \textless \ 2x + 2$. To solve this inequality, we need to isolate the variable $x$ on one side of the inequality sign. The first step is to subtract 2 from both sides of the inequality.

Subtracting 2 from Both Sides

Subtracting 2 from both sides gives us $-8 - 2 \ \textless \ 2x + 2 - 2$, which simplifies to $-10 \ \textless \ 2x$.

Dividing by 2

Next, we need to divide both sides of the inequality by 2 to solve for $x$. However, when we divide both sides by a negative number, the direction of the inequality sign changes. Since we are dividing by 2, which is positive, the direction of the inequality sign remains the same.

Solving for x

Dividing both sides by 2 gives us $\frac{-10}{2} \ \textless \ \frac{2x}{2}$, which simplifies to $-5 \ \textless \ x$.

Understanding the Solution

The solution to the inequality $-8 \ \textless \ 2x + 2$ is $x \ \textgreater \ -5$. This means that any value of $x$ greater than -5 is a solution to the inequality.

Evaluating the Options

Now that we have the solution to the inequality, we can evaluate the options given.

Option I: -5

Since $x \ \textgreater \ -5$, option I, which is -5, is not a solution to the inequality.

Option II: -10

Since $x \ \textgreater \ -5$, option II, which is -10, is also not a solution to the inequality.

Option III: -4

Since $x \ \textgreater \ -5$, option III, which is -4, is a solution to the inequality.

Conclusion

Based on the solution to the inequality, the correct answer is D. III only.

Additional Examples

To further illustrate the concept, let's consider a few more examples.

Example 1

Solve the inequality $3x - 2 \ \textless \ 5$.

Step 1: Add 2 to Both Sides

Adding 2 to both sides gives us $3x - 2 + 2 \ \textless \ 5 + 2$, which simplifies to $3x \ \textless \ 7$.

Step 2: Divide by 3

Dividing both sides by 3 gives us $\frac{3x}{3} \ \textless \ \frac{7}{3}$, which simplifies to $x \ \textless \ \frac{7}{3}$.

Step 3: Simplify the Fraction

Simplifying the fraction gives us $x \ \textless \ 2\frac{1}{3}$.

Step 4: Write the Solution

The solution to the inequality is $x \ \textless \ 2\frac{1}{3}$.

Example 2

Solve the inequality $2x + 5 \ \textless \ 11$.

Step 1: Subtract 5 from Both Sides

Subtracting 5 from both sides gives us $2x + 5 - 5 \ \textless \ 11 - 5$, which simplifies to $2x \ \textless \ 6$.

Step 2: Divide by 2

Dividing both sides by 2 gives us $\frac{2x}{2} \ \textless \ \frac{6}{2}$, which simplifies to $x \ \textless \ 3$.

Step 3: Write the Solution

The solution to the inequality is $x \ \textless \ 3$.

Conclusion

In conclusion, solving inequalities involves isolating the variable on one side of the inequality sign and following the rules of inequality signs when dividing or multiplying both sides by a negative number. By following these steps, we can solve inequalities and determine which values are solutions to the inequality.

Final Thoughts

Solving inequalities is an important concept in mathematics, and it has many real-world applications. By understanding how to solve inequalities, we can make informed decisions and solve problems in a variety of fields, including science, engineering, and economics.

Frequently Asked Questions

Q: What is an inequality?

A: An inequality is a mathematical statement that compares two expressions using a symbol such as <, >, ≤, or ≥.

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 adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

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, where a, b, and c are constants. A quadratic inequality is an inequality that can be written in the form ax^2 + bx + c < d, where a, b, and c are constants.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to factor the quadratic expression and then use the sign of the quadratic expression to determine the solution set.

Q: What is the solution set of an inequality?

A: The solution set of an inequality is the set of all values of the variable that satisfy the inequality.

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

A: To graph the solution set of an inequality, you need to plot the boundary line of the inequality and then shade the region that satisfies the inequality.

Q: What is the boundary line of an inequality?

A: The boundary line of an inequality is the line that separates the region that satisfies the inequality from the region that does not satisfy the inequality.

Q: How do I determine the direction of the inequality sign when dividing or multiplying both sides by a negative number?

A: When dividing or multiplying both sides of an inequality by a negative number, you need to reverse the direction of the inequality sign.

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

A: A strict inequality is an inequality that is written with a strict inequality sign (< or >), while a non-strict inequality is an inequality that is written with a non-strict inequality sign (≤ or ≥).

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

A: To solve a system of linear inequalities, you need to find the intersection of the solution sets of each inequality.

Q: What is the intersection of the solution sets of two inequalities?

A: The intersection of the solution sets of two inequalities is the set of all values that satisfy both inequalities.

Q: How do I graph the solution set of a system of linear inequalities?

A: To graph the solution set of a system of linear inequalities, you need to plot the boundary lines of each inequality and then shade the region that satisfies both inequalities.

Q: What is the boundary line of a system of linear inequalities?

A: The boundary line of a system of linear inequalities is the line that separates the region that satisfies both inequalities from the region that does not satisfy both inequalities.

Q: How do I determine the direction of the inequality sign when dividing or multiplying both sides of a system of linear inequalities by a negative number?

A: When dividing or multiplying both sides of a system of linear inequalities by a negative number, you need to reverse the direction of the inequality sign for each 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 objective function subject to a set of linear constraints, while a quadratic programming problem is a problem that involves maximizing or minimizing a quadratic objective function subject to a set of linear constraints.

Q: How do I solve a linear programming problem?

A: To solve a linear programming problem, you need to use a linear programming algorithm such as the simplex method or the interior-point method.

Q: What is the simplex method?

A: The simplex method is a linear programming algorithm that uses a sequence of linear programming problems to find the optimal solution to a linear programming problem.

Q: What is the interior-point method?

A: The interior-point method is a linear programming algorithm that uses a sequence of linear programming problems to find the optimal solution to a linear programming problem.

Q: How do I solve a quadratic programming problem?

A: To solve a quadratic programming problem, you need to use a quadratic programming algorithm such as the quadratic programming algorithm or the interior-point method.

Q: What is the quadratic programming algorithm?

A: The quadratic programming algorithm is a quadratic programming algorithm that uses a sequence of quadratic programming problems to find the optimal solution to a quadratic programming problem.

Q: What is the difference between a convex optimization problem and a non-convex optimization problem?

A: A convex optimization problem is a problem that involves maximizing or minimizing a convex function subject to a set of convex constraints, while a non-convex optimization problem is a problem that involves maximizing or minimizing a non-convex function subject to a set of convex constraints.

Q: How do I solve a convex optimization problem?

A: To solve a convex optimization problem, you need to use a convex optimization algorithm such as the interior-point method or the gradient descent method.

Q: What is the interior-point method?

A: The interior-point method is a convex optimization algorithm that uses a sequence of convex optimization problems to find the optimal solution to a convex optimization problem.

Q: What is the gradient descent method?

A: The gradient descent method is a convex optimization algorithm that uses a sequence of convex optimization problems to find the optimal solution to a convex optimization problem.

Q: How do I solve a non-convex optimization problem?

A: To solve a non-convex optimization problem, you need to use a non-convex optimization algorithm such as the genetic algorithm or the simulated annealing algorithm.

Q: What is the genetic algorithm?

A: The genetic algorithm is a non-convex optimization algorithm that uses a sequence of non-convex optimization problems to find the optimal solution to a non-convex optimization problem.

Q: What is the simulated annealing algorithm?

A: The simulated annealing algorithm is a non-convex optimization algorithm that uses a sequence of non-convex optimization problems to find the optimal solution to a non-convex optimization problem.

Q: What is the difference between a deterministic algorithm and a stochastic algorithm?

A: A deterministic algorithm is an algorithm that always produces the same output for a given input, while a stochastic algorithm is an algorithm that produces a random output for a given input.

Q: How do I determine the convergence of an algorithm?

A: To determine the convergence of an algorithm, you need to check if the algorithm produces the same output for a given input over a sequence of iterations.

Q: What is the difference between a local minimum and a global minimum?

A: A local minimum is a minimum value of a function that is achieved at a point in the neighborhood of the point, while a global minimum is a minimum value of a function that is achieved at a point in the entire domain of the function.

Q: How do I determine the local minimum of a function?

A: To determine the local minimum of a function, you need to use a local optimization algorithm such as the gradient descent method or the Newton's method.

Q: What is the gradient descent method?

A: The gradient descent method is a local optimization algorithm that uses a sequence of local optimization problems to find the local minimum of a function.

Q: What is Newton's method?

A: Newton's method is a local optimization algorithm that uses a sequence of local optimization problems to find the local minimum of a function.

Q: How do I determine the global minimum of a function?

A: To determine the global minimum of a function, you need to use a global optimization algorithm such as the genetic algorithm or the simulated annealing algorithm.

Q: What is the genetic algorithm?

A: The genetic algorithm is a global optimization algorithm that uses a sequence of global optimization problems to find the global minimum of a function.

Q: What is the simulated annealing algorithm?

A: The simulated annealing algorithm is a global optimization algorithm that uses a sequence of global optimization problems to find the global minimum of a function.

Q: What is the difference between a convex function and a non-convex function?

A: A convex function is a function that is always increasing or decreasing in the neighborhood of a point, while a non-convex function is a function that is not always increasing or decreasing in the neighborhood of a point.

Q: How do I determine the convexity of a function?

A: To determine the convexity of a function, you need to check if the function is always increasing or decreasing in the neighborhood of a point.

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

A: A linear function is a function that can be written in the form ax + b, where a and b are constants, while a quadratic function is a function that can be written in the form ax^2 + bx + c, where a, b, and c are constants.

Q: How do I determine the linearity of a function?

A: To determine the linearity of a function, you need to check if the function can be written in the form ax + b, where a and b are constants.

Q: What is the difference between a quadratic function and a cubic function?

A: A quadratic function is a function that can be written in the form ax^2 + bx + c, where a, b, and c are constants, while a cubic function is a function that can be written in the form ax^3 + bx^2 + cx + d, where a, b, c, and d are constants.

Q: How do I determine the quadraticity of a function?

A: To determine the quadraticity of a