Exercise 6.3Work Out These:1. { (-4 \times -3) - (-2 \times 1)$}$2. { (-15 \div 2) - (4 \times -6)$}$3. ${$4 - 4 + 3 + 2 + 3 + 4 - 5 - 6 - 9 + 1$}$4. { \frac{-2 + 12}{-5}$} 5. \[ 5. \[ 5. \[ \frac{-4x - 3}{-4 +

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Introduction

Mathematical expressions are a fundamental part of mathematics, and solving them is a crucial skill for students to master. In this article, we will work through five mathematical expressions, using a step-by-step approach to simplify and solve each one. We will cover a range of mathematical operations, including multiplication, division, addition, and subtraction, as well as fractions and algebraic expressions.

Exercise 1: Simplifying a Mathematical Expression

Problem Statement

The first expression we will work through is:

(βˆ’4Γ—βˆ’3)βˆ’(βˆ’2Γ—1)(-4 \times -3) - (-2 \times 1)

Solution

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

  1. Multiply -4 and -3: βˆ’4Γ—βˆ’3=12-4 \times -3 = 12
  2. Multiply -2 and 1: βˆ’2Γ—1=βˆ’2-2 \times 1 = -2
  3. Subtract -2 from 12: 12βˆ’(βˆ’2)=12+2=1412 - (-2) = 12 + 2 = 14

Therefore, the simplified expression is:

1414

Explanation

In this expression, we first multiplied -4 and -3, which resulted in a positive number. We then multiplied -2 and 1, which resulted in a negative number. Finally, we subtracted the negative number from the positive number, resulting in a positive number.

Exercise 2: Simplifying a Mathematical Expression

Problem Statement

The second expression we will work through is:

(βˆ’15Γ·2)βˆ’(4Γ—βˆ’6)(-15 \div 2) - (4 \times -6)

Solution

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

  1. Divide -15 by 2: βˆ’15Γ·2=βˆ’7.5-15 \div 2 = -7.5
  2. Multiply 4 and -6: 4Γ—βˆ’6=βˆ’244 \times -6 = -24
  3. Subtract -24 from -7.5: βˆ’7.5βˆ’(βˆ’24)=βˆ’7.5+24=16.5-7.5 - (-24) = -7.5 + 24 = 16.5

Therefore, the simplified expression is:

16.516.5

Explanation

In this expression, we first divided -15 by 2, resulting in a negative number. We then multiplied 4 and -6, resulting in a negative number. Finally, we subtracted the negative number from the negative number, resulting in a positive number.

Exercise 3: Simplifying a Mathematical Expression

Problem Statement

The third expression we will work through is:

4βˆ’4+3+2+3+4βˆ’5βˆ’6βˆ’9+14 - 4 + 3 + 2 + 3 + 4 - 5 - 6 - 9 + 1

Solution

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

  1. Subtract 4 from 4: 4βˆ’4=04 - 4 = 0
  2. Add 3 to 0: 0+3=30 + 3 = 3
  3. Add 2 to 3: 3+2=53 + 2 = 5
  4. Add 3 to 5: 5+3=85 + 3 = 8
  5. Add 4 to 8: 8+4=128 + 4 = 12
  6. Subtract 5 from 12: 12βˆ’5=712 - 5 = 7
  7. Subtract 6 from 7: 7βˆ’6=17 - 6 = 1
  8. Subtract 9 from 1: 1βˆ’9=βˆ’81 - 9 = -8
  9. Add 1 to -8: βˆ’8+1=βˆ’7-8 + 1 = -7

Therefore, the simplified expression is:

βˆ’7-7

Explanation

In this expression, we first subtracted 4 from 4, resulting in 0. We then added 3 to 0, resulting in 3. We continued to add and subtract numbers, following the order of operations, until we arrived at the final answer of -7.

Exercise 4: Simplifying a Fraction

Problem Statement

The fourth expression we will work through is:

βˆ’2+12βˆ’5\frac{-2 + 12}{-5}

Solution

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

  1. Add -2 and 12: βˆ’2+12=10-2 + 12 = 10
  2. Divide 10 by -5: 10βˆ’5=βˆ’2\frac{10}{-5} = -2

Therefore, the simplified expression is:

βˆ’2-2

Explanation

In this expression, we first added -2 and 12, resulting in 10. We then divided 10 by -5, resulting in -2.

Exercise 5: Simplifying an Algebraic Expression

Problem Statement

The fifth expression we will work through is:

βˆ’4xβˆ’3βˆ’4+x\frac{-4x - 3}{-4 + x}

Solution

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

  1. Factor out -4 from the numerator: βˆ’4xβˆ’3=βˆ’4(x+34)-4x - 3 = -4(x + \frac{3}{4})
  2. Factor out -4 from the denominator: βˆ’4+x=βˆ’(4βˆ’x)-4 + x = -(4 - x)
  3. Simplify the expression: βˆ’4(x+34)βˆ’(4βˆ’x)=4(x+34)4βˆ’x\frac{-4(x + \frac{3}{4})}{-(4 - x)} = \frac{4(x + \frac{3}{4})}{4 - x}

Therefore, the simplified expression is:

4(x+34)4βˆ’x\frac{4(x + \frac{3}{4})}{4 - x}

Explanation

In this expression, we first factored out -4 from the numerator and denominator, resulting in a simplified expression.

Conclusion

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

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. PEMDAS stands for:

  • Parentheses: Evaluate expressions inside parentheses first.
  • Exponents: Evaluate any exponential expressions next.
  • Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  • Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a mathematical expression?

A: To simplify a mathematical expression, follow these steps:

  1. Evaluate any expressions inside parentheses.
  2. Evaluate any exponential expressions.
  3. Evaluate any multiplication and division operations from left to right.
  4. Evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a fraction and a decimal?

A: A fraction is a way of expressing a part of a whole, where the numerator (top number) is divided by the denominator (bottom number). For example, 1/2 is a fraction. A decimal is a way of expressing a fraction as a number with a point (.) separating the whole number part from the fractional part. For example, 0.5 is a decimal.

Q: How do I add and subtract fractions?

A: To add and subtract fractions, follow these steps:

  1. Make sure the denominators (bottom numbers) are the same.
  2. Add or subtract the numerators (top numbers).
  3. Keep the same denominator.

For example, to add 1/4 and 1/4, you would get:

1/4 + 1/4 = 2/4 = 1/2

Q: How do I multiply and divide fractions?

A: To multiply and divide fractions, follow these steps:

  1. Multiply or divide the numerators (top numbers).
  2. Multiply or divide the denominators (bottom numbers).
  3. Simplify the resulting fraction.

For example, to multiply 1/2 and 1/3, you would get:

1/2 Γ— 1/3 = 1/6

Q: What is the difference between a variable and a constant?

A: A variable is a letter or symbol that represents a value that can change. For example, x is a variable. A constant is a value that does not change. For example, 5 is a constant.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, follow these steps:

  1. Combine like terms (terms with the same variable).
  2. Simplify any fractions.
  3. Simplify any exponential expressions.

For example, to simplify the expression 2x + 3x, you would get:

2x + 3x = 5x

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

A: A linear equation is an equation with one variable and a linear relationship between the variables. For example, 2x + 3 = 5 is a linear equation. A quadratic equation is an equation with one variable and a quadratic relationship between the variables. For example, x^2 + 2x + 1 = 0 is a quadratic equation.

Conclusion

In this article, we answered some frequently asked questions about mathematical expressions. We covered topics such as the order of operations, simplifying expressions, adding and subtracting fractions, multiplying and dividing fractions, variables and constants, and linear and quadratic equations. By following these steps and understanding these concepts, you can simplify and solve mathematical expressions with ease.