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Chapter 2: Problem 102

Solve. Label any contradictions or identities. $$[7-2(8 \div(-2))] x=0$$

Short Answer

Expert verified
x = 0.

Step by step solution

01

Simplify the expression inside the parentheses

First, simplify the expression inside the parentheses. Compute the division: $$8 \text { divided by } (-2) = -4.$$ Then, substitute back into the expression: $$7 - 2(-4).$$
02

Simplify the Expression Further

Now, compute the multiplication and subtraction: $$2 \times(-4) = -8.$$ Then $$7 - (-8) = 7 + 8 = 15.$$ So, the final expression is $$15 x = 0.$$
03

Solve for x

To find the value of x, divide both sides of the equation by 15: $$x=\frac{0}{15}=0.$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

simplifying expressions
Simplifying expressions is an essential skill in algebra. It helps break down complex problems into more manageable parts. To simplify an expression, follow the order of operations: parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right). Always look for opportunities to combine like terms and simplify step by step. For example, in the given problem \([7-2(8 \div(-2))] x=0\), we start by handling the division inside the parentheses first.
division
Division is one of the fundamental operations in algebra. When dividing numbers, you determine how many times one number is contained within another. In our problem, we need to divide 8 by -2. The result is -4 because dividing a positive number by a negative number yields a negative result. So, 8 divided by -2 equals -4. This intermediate step is crucial for the next operations in the expression simplifying process.
multiplication
Multiplication is another core algebraic operation. In the context of our problem, after dividing inside the parentheses and getting -4, we need to multiply -4 by -2. Remember that multiplying two negative numbers results in a positive number. Thus, -2 multiplied by -4 is 8. This step transforms the expression to something more workable: 7 - (-8).
identities in algebra
'Identities in algebra' refer to equations that are true for all possible values of the variable(s) involved. The given exercise simplifies to 15x = 0 after all intermediate steps are performed. Here, we solve for x by dividing both sides by 15. Mathematically: \( x = \frac{0}{15}=0 \). An identity shows us that x = 0 is always true regardless of other factors, which signifies that for this problem, zero is the only possible solution. Understanding identities helps in quickly identifying solutions and contradictions in algebraic equations.

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Most popular questions from this chapter

Use an inequality and the five-step process to solve each problem. The U.S. Postal Service defines a "package" as a parcel for which the sum of the length and the girth is less than 84 in. (Length is the longest side of a package and girth is the distance around the other two sides of the package.) A box has a fixed girth of 29 in. Determine (in terms of an inequality) those lengths for which the box is considered a "package." (THE IMAGE CANNOT BE COPY)

The perimeter of a rectangle is \(101.74 \mathrm{cm} .\) If the length is \(4.25 \mathrm{cm}\) longer than the width, find the dimensions of the rectangle.

To prepare for Section 2.5, review translating to algebraic expressions and equations. Translate to an algebraic expression or equation. It has been determined that at the age of \(15,\) a boy has reached \(96.1 \%\) of his final adult height. Jaraan is \(6 \mathrm{ft} 4\) in. at the age of \(15 .\) What will his final adult height be?

Use an inequality and the five-step process to solve each problem. Rod's quiz grades are \(73,75,89,\) and 91. What scores on a fifth quiz will make his average quiz grade at least \(85 ?\)

Translate to an inequality. The cost of gasoline was at most \(\$ 4\) per gallon.
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