Problem 127 - Introductory Algebra for College Students

Question

Solve each inequality. Use a calculator to help with the arithmetic. \(1.45-7.23 x>-1.442\)

Step-by-Step Solution

Answer

The solution to the inequality is all real numbers.

Step 1: Isolate the 'x' term

To make 'x' by itself on one side of the inequality, add '7.23x' to both sides, so the inequality becomes \(1.45+7.23x > -1.442+7.23x\). After doing the addition, the inequality simplifies to \(1.45 > -1.442 \)

Step 2: Solve the inequality

Since there is no 'x' on the right side, continue the operation with \(1.45 > -1.442\). Clearly, 1.45 is greater than -1.442.

Step 3: Determine the solution

As there is no 'x' left in the inequality after the arithmetic operation from Step 2, and since 1.45 is indeed greater than -1.442, it means that the original inequality is true for all real numbers. Hence, the solution to the inequality \(1.45 - 7.23x > -1.442\) is all real numbers.

Key Concepts

algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In inequalities like the one in the exercise, algebra helps to rearrange and simplify the expression by using arithmetic operations such as addition and subtraction.
In the given problem, the goal is to isolate the variable 'x'. This involves performing operations on both sides of the inequality to get 'x' by itself.
For example, in this exercise, we added '7.23x' to both sides to remove it from the side containing other numbers, eventually simplifying the inequality.

• First, focus on rearranging the inequality to express it in a simpler form.
• Always aim to isolate 'x' to understand what values it can take.
• This helps indicate the range of possible solutions.

Algebraic manipulation is key to solving these inequalities, allowing us to determine if the relationship holds for some or all real numbers.

real numbers

Real numbers are any numbers that can be found on the number line. This includes both positive and negative numbers, decimal numbers, and zero.
In the context of inequalities like this one, understanding real numbers is crucial because the solution might encompass a range or all possible values.
For this exercise, after simplifying, we observe that 1.45 is indeed greater than -1.442 when we compare regular numbers.

• The fact that no 'x' remains upon solving means that the inequality holds for all real numbers.
• This implies that any real number can be a valid solution.

The concept of real numbers helps us understand the breadth of potential solutions in algebra and calculus.

calculator usage

Using a calculator can significantly simplify the arithmetic involved in solving inequalities. It helps to ensure accuracy in operations, especially when dealing with decimals or large numbers.
For instance, in this problem, a calculator aids in quickly verifying that 1.45 is indeed greater than -1.442.
Calculators make it easier to spot arithmetic errors, which can lead to mistakes in solving the inequality.

• Utilize calculators for accurate decimal and whole number arithmetic.
• They are particularly helpful in cross-checking manual calculations.
• This ensures the steps taken to solve an inequality are both correct and efficient.

Despite their usefulness, it's important to first understand the underlying math concepts, so tools are used appropriately and effectively.