1. Evaluate The Expression: Evaluate The Following Expression Without Using A Calculator: $\[ \frac{-2(5+3)-9 \div 3+5}{-3x-5-2 \times 4} \\]2. Oranges Sales Problem: A Vendor Bought 1948 Oranges On A Thursday And Sold 750 Of Them

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Introduction

In this article, we will evaluate the given mathematical expression without using a calculator. The expression is a combination of arithmetic operations, including addition, subtraction, multiplication, and division. We will break down the expression into smaller parts and solve each part step by step.

Evaluate the Expression

The given expression is:

−2(5+3)−9÷3+5−3x−5−2×4{ \frac{-2(5+3)-9 \div 3+5}{-3x-5-2 \times 4} }

To evaluate this expression, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate the expressions inside the parentheses.
  2. Exponents: Evaluate any exponential expressions.
  3. Multiplication and Division: Evaluate the multiplication and division operations from left to right.
  4. Addition and Subtraction: Evaluate the addition and subtraction operations from left to right.

Step 1: Evaluate the Expressions Inside the Parentheses

The expression inside the parentheses is:

−2(5+3)-2(5+3)

To evaluate this expression, we need to follow the order of operations:

  1. Evaluate the expression inside the parentheses: 5+3=85+3 = 8
  2. Multiply −2-2 by the result: −2×8=−16-2 \times 8 = -16

Step 3: Evaluate the Division Operation

The expression is:

−9÷3-9 \div 3

To evaluate this expression, we need to divide −9-9 by 33:

−9÷3=−3-9 \div 3 = -3

Step 4: Evaluate the Multiplication Operation

The expression is:

−2×4-2 \times 4

To evaluate this expression, we need to multiply −2-2 by 44:

−2×4=−8-2 \times 4 = -8

Step 5: Substitute the Values Back into the Original Expression

Now that we have evaluated the expressions inside the parentheses, division, and multiplication, we can substitute the values back into the original expression:

−16−(−3)+5−3x−5−(−8){ \frac{-16-(-3)+5}{-3x-5-(-8)} }

Step 6: Simplify the Expression

To simplify the expression, we need to follow the order of operations:

  1. Evaluate the expressions inside the parentheses: −3+5=2-3+5 = 2
  2. Substitute the value back into the expression: ${ \frac{-16+2}{-3x-5+8} }$
  3. Simplify the expression: ${ \frac{-14}{-3x+3} }$

Step 7: Solve for x

To solve for x, we need to isolate x on one side of the equation. We can do this by multiplying both sides of the equation by −3x+3-3x+3:

−14=−3x2+3x-14 = -3x^2+3x

Step 8: Rearrange the Equation

To rearrange the equation, we need to move all the terms to one side of the equation:

−3x2+3x+14=0-3x^2+3x+14 = 0

Step 9: Solve the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2-4ac}}{2a}

In this case, a=−3a = -3, b=3b = 3, and c=14c = 14. Plugging these values into the formula, we get:

x=−3±32−4(−3)(14)2(−3)x = \frac{-3 \pm \sqrt{3^2-4(-3)(14)}}{2(-3)}

x=−3±9+168−6x = \frac{-3 \pm \sqrt{9+168}}{-6}

x=−3±177−6x = \frac{-3 \pm \sqrt{177}}{-6}

Conclusion

In this article, we evaluated the given mathematical expression without using a calculator. We broke down the expression into smaller parts and solved each part step by step. We used the order of operations (PEMDAS) to evaluate the expressions inside the parentheses, division, and multiplication. We then substituted the values back into the original expression and simplified it. Finally, we solved for x by using the quadratic formula.

Oranges Sales Problem

Introduction

A vendor bought 1948 oranges on a Thursday and sold 750 of them. The vendor wants to know how many oranges are left. We will use the concept of subtraction to solve this problem.

Step 1: Identify the Number of Oranges Bought

The vendor bought 1948 oranges.

Step 2: Identify the Number of Oranges Sold

The vendor sold 750 oranges.

Step 3: Subtract the Number of Oranges Sold from the Number of Oranges Bought

To find the number of oranges left, we need to subtract the number of oranges sold from the number of oranges bought:

1948 - 750 = 1198

Conclusion

In this article, we solved the oranges sales problem by using the concept of subtraction. We identified the number of oranges bought and sold, and then subtracted the number of oranges sold from the number of oranges bought to find the number of oranges left.

Discussion

  • What is the order of operations (PEMDAS)?
  • How do you evaluate expressions inside parentheses?
  • How do you evaluate division and multiplication operations?
  • How do you simplify an expression?
  • How do you solve a quadratic equation?

Final Answer

The final answer is 1198\boxed{1198}.

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Introduction

In this article, we will answer some frequently asked questions about evaluating expressions and solving problems. We will cover topics such as the order of operations, evaluating expressions inside parentheses, and solving quadratic equations.

Q&A

Q: What is the order of operations (PEMDAS)?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. PEMDAS stands for Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction. We need to follow this order to evaluate expressions correctly.

Q: How do you evaluate expressions inside parentheses?

A: To evaluate expressions inside parentheses, we need to follow the order of operations. We need to evaluate any exponential expressions first, then any multiplication and division operations from left to right, and finally any addition and subtraction operations from left to right.

Q: How do you evaluate division and multiplication operations?

A: To evaluate division and multiplication operations, we need to follow the order of operations. We need to perform the operations from left to right. For example, if we have the expression 12 ÷ 3 × 2, we need to perform the division operation first, then the multiplication operation.

Q: How do you simplify an expression?

A: To simplify an expression, we need to follow the order of operations and combine like terms. We need to evaluate any exponential expressions first, then any multiplication and division operations from left to right, and finally any addition and subtraction operations from left to right.

Q: How do you solve a quadratic equation?

A: To solve a quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

We need to plug in the values of a, b, and c into the formula and simplify the expression.

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 = 0 is a linear equation. A quadratic equation 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 you graph a linear equation?

A: To graph a linear equation, we need to find two points on the line and plot them on a coordinate plane. We can then draw a line through the two points to represent the equation.

Q: How do you graph a quadratic equation?

A: To graph a quadratic equation, we need to find the vertex of the parabola and plot it on a coordinate plane. We can then draw a parabola through the vertex to represent the equation.

Conclusion

In this article, we answered some frequently asked questions about evaluating expressions and solving problems. We covered topics such as the order of operations, evaluating expressions inside parentheses, and solving quadratic equations. We hope that this article has been helpful in answering your questions and providing you with a better understanding of these concepts.

Discussion

  • What is the order of operations (PEMDAS)?
  • How do you evaluate expressions inside parentheses?
  • How do you evaluate division and multiplication operations?
  • How do you simplify an expression?
  • How do you solve a quadratic equation?
  • What is the difference between a linear equation and a quadratic equation?
  • How do you graph a linear equation?
  • How do you graph a quadratic equation?

Final Answer

There is no final answer to this article, as it is a Q&A article. However, we hope that the answers to the questions have been helpful in providing you with a better understanding of the concepts.