Q4: Solve These Equationsa. { \frac 4}{1-x} = \frac{3}{2x}$}$b. ${ 3(2x-1) - 5 = X + 7\$} C. ${ 15 - (x-3) = 6-x\$} ---Q5 Solve These Inequalitiesa. ${$4(x+4) - 3(x-7) \ \textless \ 4x - 7$ $b.

Introduction

In mathematics, equations and inequalities are fundamental concepts that help us solve problems and understand relationships between variables. In this article, we will focus on solving equations and inequalities, specifically in the context of Q4. We will break down each problem step by step, using algebraic techniques to find the solutions.

Solving Equations

a. Solving the Equation $\frac{4}{1-x} = \frac{3}{2x}$

To solve this equation, we can start by cross-multiplying both sides:

$$4(2x) = 3(1-x)$$

Expanding both sides, we get:

$$8x = 3 - 3x$$

Adding $3x$ to both sides, we get:

$$11x = 3$$

Dividing both sides by $11$, we get:

$$x = \frac{3}{11}$$

Therefore, the solution to the equation is $x = \frac{3}{11}$.

b. Solving the Equation $3(2x-1) - 5 = x + 7$

To solve this equation, we can start by expanding the left-hand side:

$$6x - 3 - 5 = x + 7$$

Simplifying the left-hand side, we get:

$$6x - 8 = x + 7$$

Subtracting $x$ from both sides, we get:

$$5x - 8 = 7$$

Adding $8$ to both sides, we get:

$$5x = 15$$

Dividing both sides by $5$, we get:

$$x = 3$$

Therefore, the solution to the equation is $x = 3$.

c. Solving the Equation $15 - (x-3) = 6-x$

To solve this equation, we can start by simplifying the left-hand side:

$$15 - x + 3 = 6 - x$$

Simplifying further, we get:

$$18 - x = 6 - x$$

Adding $x$ to both sides, we get:

$$18 = 6$$

This is a contradiction, which means that there is no solution to the equation.

Solving Inequalities

a. Solving the Inequality $4(x+4) - 3(x-7) < 4x - 7$

To solve this inequality, we can start by expanding both sides:

$$4x + 16 - 3x + 21 < 4x - 7$$

Simplifying both sides, we get:

$$x + 37 < 4x - 7$$

Subtracting $x$ from both sides, we get:

$$37 < 3x - 7$$

Adding $7$ to both sides, we get:

$$44 < 3x$$

Dividing both sides by $3$, we get:

$$\frac{44}{3} < x$$

Therefore, the solution to the inequality is $x > \frac{44}{3}$.

b. Solving the Inequality $2(x-3) + 5 > 3(x+2)$

To solve this inequality, we can start by expanding both sides:

$$2x - 6 + 5 > 3x + 6$$

Simplifying both sides, we get:

$$2x - 1 > 3x + 6$$

Subtracting $2x$ from both sides, we get:

$$-1 > x + 6$$

Subtracting $6$ from both sides, we get:

$$-7 > x$$

Therefore, the solution to the inequality is $x < -7$.

Conclusion

In this article, we have solved several equations and inequalities, using algebraic techniques to find the solutions. We have seen how to use cross-multiplication, expansion, and simplification to solve equations, and how to use inequality properties to solve inequalities. By following these techniques, we can solve a wide range of mathematical problems and understand the relationships between variables.

Key Takeaways

• To solve an equation, we can use cross-multiplication, expansion, and simplification to find the solution.
• To solve an inequality, we can use inequality properties to find the solution.
• Algebraic techniques are essential for solving mathematical problems and understanding relationships between variables.

Further Reading

For further reading on solving equations and inequalities, we recommend the following resources:

• Algebraic Techniques for Solving Equations and Inequalities
• Solving Equations and Inequalities
• Mathematics for Dummies

Discussion

Introduction

In our previous article, we covered the basics of solving equations and inequalities. However, we know that there are many more questions and concerns that you may have. In this article, we will address some of the most frequently asked questions about solving equations and inequalities.

Q: What is the difference between an equation and an inequality?

A: An equation is a statement that says two expressions are equal, while an inequality is a statement that says one expression is greater than or less than another expression.

Q: How do I solve a linear equation?

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

1. Simplify the equation by combining like terms.
2. Isolate the variable by adding or subtracting the same value to both sides of the equation.
3. Divide both sides of the equation by the coefficient of the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

1. Factor the equation, if possible.
2. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
3. Simplify the equation and solve for x.

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, c, and d are constants.

Q: How do I solve a linear inequality?

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

1. Simplify the inequality by combining like terms.
2. Isolate the variable by adding or subtracting the same value to both sides of the inequality.
3. Divide both sides of the inequality by the coefficient of the variable.

Q: How do I solve a quadratic inequality?

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

1. Factor the inequality, if possible.
2. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
3. Simplify the inequality and solve for x.

Q: What is the difference between a rational inequality and a radical inequality?

A: A rational inequality is an inequality that can be written in the form ax + b < c, where a, b, and c are constants and a is not equal to 0. A radical inequality is an inequality that can be written in the form √x < y, where x and y are constants.

Q: How do I solve a rational inequality?

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

1. Simplify the inequality by combining like terms.
2. Isolate the variable by adding or subtracting the same value to both sides of the inequality.
3. Divide both sides of the inequality by the coefficient of the variable.

Q: How do I solve a radical inequality?

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

1. Simplify the inequality by combining like terms.
2. Isolate the variable by adding or subtracting the same value to both sides of the inequality.
3. Square both sides of the inequality to eliminate the radical.

Conclusion

We hope that this article has been helpful in answering some of the most frequently asked questions about solving equations and inequalities. Remember to always follow the steps outlined above and to simplify the equation or inequality as much as possible before solving for the variable.

Key Takeaways

• To solve an equation, you can use cross-multiplication, expansion, and simplification to find the solution.
• To solve an inequality, you can use inequality properties to find the solution.
• Algebraic techniques are essential for solving mathematical problems and understanding relationships between variables.

Further Reading

For further reading on solving equations and inequalities, we recommend the following resources:

• Algebraic Techniques for Solving Equations and Inequalities
• Solving Equations and Inequalities
• Mathematics for Dummies

Discussion

We hope that this article has been helpful in understanding how to solve equations and inequalities. If you have any questions or comments, please feel free to discuss them below.