Problem 54 - Elementary Algebra

Question

Determine whether the statements for the following problems are true or false. $$6-4(4)(1) \leq 10$$

Step-by-Step Solution

Answer

Answer: False.

Step 1: Evaluate the expression on the left side of the inequality

We will follow the order of operations (PEMDAS/BODMAS) to evaluate the left side: 1. Parentheses/Brackets: There are no brackets in the expression. 2. Exponents/Orders: There are no exponents in the expression. 3. Multiplication and Division: $$-4 \times 4 \times 1 = -16$$ 4. Addition and subtraction: $$6 - (-16) = 6 + 16 = 22$$ So, the left side of the inequality simplifies to 22.

Step 2: Compare the calculated result to the right side of the inequality

Now that we have the left side expression evaluated, we will compare it to the right side which is 10. The inequality is written as: $$22 \leq 10$$

Step 3: Determine if the inequality is true or false

By looking at the numbers, we can see that 22 is greater than 10, not less than or equal to. Thus, the inequality is false.

Key Concepts

Order of Operations

Understanding the order of operations is crucial when working with mathematical expressions, especially when they're part of an inequality problem. This set of rules tells us the sequence in which we should solve different parts of an arithmetic expression to achieve the correct result.

We often utilize the acronym PEMDAS in the United States, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Another common acronym is BODMAS, used in other countries, standing for Brackets, Orders (another term for exponents), Division, Multiplication, Addition, and Subtraction. Both acronyms essentially refer to the same set of operations in the order they should be performed.

Let's break it down with an example problem from the textbook solutions:
For the expression $6\; -\; 4(4)(1)$

• We first look for parentheses or brackets but there are none.
• Next, we check for exponents, which also do not appear in this expression.
• We then proceed with multiplication and division from left to right. In our case, we multiply $-4,\; 4,$ and $1,$ which gives us $-16.$
• Lastly, we tackle addition and subtraction from left to right. Thus, we add $6$ to $-16$, resulting in $22.$

By following the proper order of operations, we correctly evaluated the left side of the inequality, ensuring that the problem is solved accurately.

Evaluating Expressions

Evaluating expressions is all about simplifying a mathematical phrase into a single number or a simpler expression. It involves performing all the arithmetic operations according to the order of operations we just discussed. When working with inequalities, accurate evaluation is vital because the truth of the inequality depends on the correct values of the expressions on either side.

In the given example, $6-4(4)(1),$ we carefully evaluated the expression step by step. Without brackets or exponents, we strictly followed multiplication yielding $-16$, and then proceeded to add this to $6$ which resulted in $22.$

This process is fundamental in mathematics, as you'll encounter expressions that require evaluation in algebra, calculus, and even beyond. The key takeaway is to always work systematically and respect the hierarchy of operations to avoid mistakes that could lead to incorrect conclusions about inequalities or any other type of mathematical problem.

Inequality Comparison

The concept of inequality comparison is where we determine the relationship between two values or expressions. Inequalities, such as '$eq$', '$>$', or '$gtr$', signify that the two sides are not necessarily equal – one is either greater than or less than the other, or at times we state that one side is greater than or equal to '$gte$' or less than or equal to '$gleq$'. Comparing correctly is crucial for correctly solving inequality problems.

In our sample problem, after evaluating the expression, we ended up with $22$ on the left side and have the number $10$ on the right. The comparison operation in this inequality is '$gleq$', which means 'less than or equal to'. When we compare $22$ and $10,$ it's evident that $22$ is not less than or equal to $10;\; 22$ is actually greater. Thus, the statement $22\; gleq\; 10$ is false.

It's essential to be vigilant when comparing values in inequalities, as this determines the solution's validity. Being accurate in this step is fundamental for anyone looking to master mathematical problems involving inequalities.