Problem 40 - Elementary Algebra

Question

Simplify. (Assume all denominators are nonzero.) $$ 3 x 2 x-1-x-4 x+4+12(2-x) 2 x 2+7 x-4 $$

Step-by-Step Solution

Answer

The simplified expression is \(-15x + 27\).

Step 1: Distribute the Constants

Begin by distributing the constants in the problem, specifically the number 12 in the expression \(12(2 - x)\). Utilize the distributive property \(a(b + c) = ab + ac\):\[12(2 - x) = 12 \times 2 - 12 \times x = 24 - 12x.\]

Step 2: Expand the Remaining Terms

Examine the remaining terms, \(3x\), \(-1\), \,\(-x\), \,\(-4x\), \,\(+4\), \,\(+24\), \,\(-12x\), as well as the other standalone terms and constants. Write them out as follows:\[3x, \quad -1, \quad -x, \quad -4x, \quad +4, \quad +24 - 12x.\]

Step 3: Combine Like Terms

Group and combine like terms, meaning combine the terms with \(x\) and the constant terms separately:- For \(x\):-\[3x - x - 4x - 12x = 3x - x - 4x - 12x = 3x - 17x = -15x.\]- For constants:\[-1 + 4 + 24 = 27.\]

Step 4: Write the Simplified Expression

Combine the results from Step 3:\[-15x + 27\] this is the simplified expression.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that helps simplify expressions by distributing a multiplication over addition or subtraction. It is expressed as \( a(b + c) = ab + ac \). This means that the multiplicative factor outside the parentheses is multiplied by each term inside the parentheses individually.

In our exercise, we see the distributive property in action in the term \( 12(2-x) \). By distributing the 12 across the terms within the parentheses, we break down the expression as follows:

• The multiplication of 12 by 2 gives us 24.
• The multiplication of 12 by \(-x\) gives us \(-12x\).

Therefore, the expression \( 12(2-x) \) simplifies to \( 24 - 12x \). Utilizing the distributive property in this way helps make complex algebraic expressions more manageable and is a critical step in the simplification process.

Combining Like Terms

Combining like terms is another essential process in simplifying expressions. It involves adding or subtracting coefficients of the same variable terms or constant terms.

After applying the distributive property, the expression in the exercise includes terms like \( 3x, -1, -x, -4x, +4, +24, \text{and} -12x \). To simplify this expression, we need to:

• Group all the terms containing \( x \) together, including \( 3x, -x, -4x, \text{and} -12x \).
• Group all the constant terms, like \(-1, +4, \text{and} +24 \).

For the terms involving \( x \), the combination is as follows:

• \( 3x - x - 4x - 12x \) becomes \( -15x \).

For the constant terms, the combination is:

• \(-1 + 4 + 24 \), which equates to 27.

By combining like terms, you can significantly reduce the complexity of an expression, making it easier to interpret and solve.

Simplifying Expressions

Simplifying expressions involves reducing an algebraic expression to its most concise form. It combines all the algebraic techniques like using the distributive property and combining like terms.

In the example given from our exercise, the simplification steps are:

• Apply the distributive property to terms like \( 12(2-x) \).
• Combine all like terms, such as those containing the same variables or constants.

Once the expression has been fully simplified, we arrive at the final expression, which in this case is:

• \(-15x + 27\)

This expression is much more straightforward than the original and communicates the relationship between \( x \) and the constants much more clearly.

Simplifying expressions is a crucial aspect of algebra, as it allows for easier manipulation and solution of equations. It requires careful application of algebraic rules and operations to ensure accuracy and simplicity in the result.