(b) Thrice A Number(c) Nine Added To One-fourth Of A Number Exceeds 27 By Five Times The Number.(d) The Product Of A Number And Some Factor(c) The Quotient Of $p$ And 7 Gives 4 More Than A Given Number.(f) Double Of $x$ Is

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Mathematical Problem-Solving: A Comprehensive Guide to Solving Equations and Inequalities

Mathematics is a subject that requires problem-solving skills, logical thinking, and analytical reasoning. In this article, we will delve into the world of mathematical problem-solving, focusing on solving equations and inequalities. We will explore various types of problems, including linear equations, quadratic equations, and inequalities, and provide step-by-step solutions to each problem.

Problem Statement

Thrice a number is 3 times the number. If 9 is added to one-fourth of a number, it exceeds 27 by five times the number.

Solution

Let's denote the number as x. We can start by translating the problem statement into an equation.

Thrice a number = 3x One-fourth of a number = x/4 Nine added to one-fourth of a number = x/4 + 9 Exceeds 27 by five times the number = 27 + 5x

Now, we can set up an equation based on the problem statement:

x/4 + 9 = 27 + 5x

To solve for x, we can start by multiplying both sides of the equation by 4 to eliminate the fraction:

x + 36 = 108 + 20x

Next, we can subtract x from both sides of the equation to get:

36 = 108 + 19x

Now, we can subtract 108 from both sides of the equation to get:

-72 = 19x

Finally, we can divide both sides of the equation by 19 to solve for x:

x = -72/19 x = -3.7895 (approximately)

Problem Statement

Nine added to one-fourth of a number exceeds 27 by five times the number.

Solution

Let's denote the number as x. We can start by translating the problem statement into an equation.

One-fourth of a number = x/4 Nine added to one-fourth of a number = x/4 + 9 Exceeds 27 by five times the number = 27 + 5x

Now, we can set up an equation based on the problem statement:

x/4 + 9 = 27 + 5x

To solve for x, we can start by multiplying both sides of the equation by 4 to eliminate the fraction:

x + 36 = 108 + 20x

Next, we can subtract x from both sides of the equation to get:

36 = 108 + 19x

Now, we can subtract 108 from both sides of the equation to get:

-72 = 19x

Finally, we can divide both sides of the equation by 19 to solve for x:

x = -72/19 x = -3.7895 (approximately)

Problem Statement

The product of a number and some factor is 24.

Solution

Let's denote the number as x and the factor as y. We can start by translating the problem statement into an equation.

Product of a number and some factor = xy xy = 24

To solve for x, we can divide both sides of the equation by y:

x = 24/y

Now, we can find the value of x by substituting different values of y.

For example, if y = 1, then x = 24/1 = 24

If y = 2, then x = 24/2 = 12

If y = 3, then x = 24/3 = 8

And so on.

Problem Statement

The quotient of p and 7 gives 4 more than a given number.

Solution

Let's denote the given number as x. We can start by translating the problem statement into an equation.

Quotient of p and 7 = p/7 4 more than a given number = x + 4

Now, we can set up an equation based on the problem statement:

p/7 = x + 4

To solve for p, we can start by multiplying both sides of the equation by 7 to eliminate the fraction:

p = 7x + 28

Now, we can find the value of p by substituting different values of x.

For example, if x = 1, then p = 7(1) + 28 = 35

If x = 2, then p = 7(2) + 28 = 42

If x = 3, then p = 7(3) + 28 = 49

And so on.

Problem Statement

Double of x is 2x.

Solution

Let's denote the number as x. We can start by translating the problem statement into an equation.

Double of x = 2x

Now, we can find the value of x by substituting different values of 2x.

For example, if 2x = 10, then x = 10/2 = 5

If 2x = 20, then x = 20/2 = 10

If 2x = 30, then x = 30/2 = 15

And so on.

In this article, we have explored various types of mathematical problems, including linear equations, quadratic equations, and inequalities. We have provided step-by-step solutions to each problem, using algebraic techniques to solve for the unknown variables. By following these examples, readers can develop their problem-solving skills and become proficient in solving mathematical equations and inequalities.
Mathematical Problem-Solving: A Comprehensive Guide to Solving Equations and Inequalities - Q&A

In our previous article, we explored various types of mathematical problems, including linear equations, quadratic equations, and inequalities. We provided step-by-step solutions to each problem, using algebraic techniques to solve for the unknown variables. In this article, we will answer some of the most frequently asked questions related to mathematical problem-solving.

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

A: A linear equation is an equation in which the highest power of the variable is 1. For example, 2x + 3 = 5 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a linear equation?

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

  1. Add or subtract the same value to both sides of the equation to isolate the variable.
  2. Multiply or divide both sides of the equation by the same value to eliminate the coefficient of the variable.
  3. Simplify the equation to find the value 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 quadratic expression, if possible.
  2. Use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.
  3. Simplify the equation to find the value of the variable.

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

A: An inequality is a statement that two expressions are not equal. For example, 2x + 3 > 5 is an inequality. An equation, on the other hand, is a statement that two expressions are equal. For example, 2x + 3 = 5 is an equation.

Q: How do I solve an inequality?

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

  1. Add or subtract the same value to both sides of the inequality to isolate the variable.
  2. Multiply or divide both sides of the inequality by the same value to eliminate the coefficient of the variable.
  3. Simplify the inequality to find the value of the variable.

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

A: A system of equations is a set of two or more equations that are solved simultaneously. For example, {x + 2y = 4, 3x - 2y = 5} is a system of equations. A system of inequalities, on the other hand, is a set of two or more inequalities that are solved simultaneously. For example, {x + 2y > 4, 3x - 2y < 5} is a system of inequalities.

Q: How do I solve a system of equations?

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

  1. Use the method of substitution to solve one equation for one variable.
  2. Substitute the expression for the variable into the other equation.
  3. Simplify the equation to find the value of the variable.

Q: How do I solve a system of inequalities?

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

  1. Graph the inequalities on a coordinate plane.
  2. Find the intersection of the two inequalities.
  3. Simplify the inequality to find the value of the variable.

In this article, we have answered some of the most frequently asked questions related to mathematical problem-solving. We have provided step-by-step solutions to each problem, using algebraic techniques to solve for the unknown variables. By following these examples, readers can develop their problem-solving skills and become proficient in solving mathematical equations and inequalities.