Problem 34 - Intermediate Algebra

Question

Simplify the algebraic expressions by removing parentheses and combining similar terms. $$ 4(-x-1)+3(-2 x-5)-2(x+1) $$

Step-by-Step Solution

Answer

The simplified expression is \(-12x - 21\).

Step 1: Distribute the Constants

First, apply the distributive property to each part of the expression: For the first part, distribute 4: \[4(-x - 1) = -4x - 4\]For the second part, distribute 3: \[3(-2x - 5) = -6x - 15\]For the third part, distribute -2: \[-2(x + 1) = -2x - 2\]This gives the expression \[-4x - 4 - 6x - 15 - 2x - 2.\]

Step 2: Combine Like Terms

Combine terms that have the same variable and degree.First, combine the terms with "x":\[-4x - 6x - 2x = -12x\]Next, combine the constant terms:\[-4 - 15 - 2 = -21\]This gives the simplified expression of:\[-12x - 21\].

Key Concepts

Distributive Property

The distributive property is a key concept in algebra. It helps us simplify expressions by spreading out a multiplication over terms inside parentheses. Think of it like sharing equally among groups. Here’s how it works:

• Identify the number or variable outside the parentheses (this is the one that needs to be distributed).
• Multiply this number by each term inside the parentheses.

For example, in the expression \(4(-x - 1)\), the number 4 is outside the parentheses. You need to multiply 4 by both \(-x\) and \(-1\). This results in \(-4x - 4\).
In the exercise, applying the distributive property systematically to each group you get:

• For \(4(-x - 1)\): \(-4x - 4\)
• For \(3(-2x - 5)\): \(-6x - 15\)
• For \(-2(x + 1)\): \(-2x - 2\)

This method breaks down an seemingly complex problem into bite-sized pieces, making it more manageable. Once you distribute, combining like terms becomes much simpler. This foundational step is vital because it ensures that each term is ready for the next stage in algebraic simplification.

Combining Like Terms

After distributing terms, the next step is combining like terms. This consolidates your expression into a simpler form. But what exactly are like terms?
Like terms are those that have the same variable raised to the same power, or, if there are no variables, just the numbers on their own. For example, all \(x\) terms are like terms, and all constant numbers are like terms:

• Combine terms with \(x\): Terms such as \(-4x\), \(-6x\), and \(-2x\) are all like terms because they each contain the variable \(x\).
• Combine constant terms: Numbers without variables, such as \(-4\), \(-15\), and \(-2\), are also like terms.

To combine, simply add or subtract the coefficients of the like terms, just as you would do with regular numbers. In our example:

• For the \(x\) terms: \(-4x - 6x - 2x = -12x\)
• For the constants: \(-4 - 15 - 2 = -21\)

This step reduces the clutter in your expression and makes it far easier to handle. Emphasizing combining like terms is crucial because it paves the way for reaching simpler, more understandable expressions.

Simplification of Expressions

Simplifying expressions is the ultimate goal in algebra to make calculations less complicated and more efficient. After distributing and combining like terms, you arrive at a cleaner, more straightforward expression.
For instance, from the original expression \(4(-x-1)+3(-2 x-5)-2(x+1)\), through distribution and combining like terms, you get the simplified form:\(-12x - 21\).
Why simplify? Simplifying expressions helps in identifying essential parts of a problem, which is especially useful in more complicated algebraic equations or in calculus. Furthermore, it allows you to solve equations easier and faster.

• Focus on Distributive Property: Ensures multiplication appearance is handled correctly.
• Focus on Like Terms: Simplifies multiple operands into concise numbers or variables.
• End Result: Clean expression to use for further equations or evaluations.

In summary, simplification is not just a process but a necessary skill in tackling mathematics efficiently. Mastering it provides stronger problem-solving capabilities for any algebraic task you might face.