Question

Use a calculator to solve the inequality. (Round each number in your answer to two decimal places.) \(\frac{1}{2.3 x-5.2}>3.4\)

Short answer

The solution to the inequality \(\frac{1}{2.3x - 5.2}>3.4\) is \(x<2.39\).

Step-by-step solution

Isolate the Fraction

The first step is to move everything involving \(x\) to one side and the constants to the other, starting with the constant 3.4. This is done by multiplying both sides of the inequality by \(2.3x - 5.2\), which gives \(1 > 3.4(2.3x - 5.2)\).

Distribute 3.4 on the Right Side

Next, multiply 3.4 by both terms inside the parentheses \(2.3x - 5.2\), which gives \(1>7.82x - 17.68\).

Isolate x

Now, isolate \(x\) on one side by first adding 17.68 to both sides of the inequality, which gives \(18.68> 7.82x\). Then, divide both sides by 7.82, which gives \(x<2.3887218\).

Round the Solution

To fulfill the requirement of the problem, round the answer to two decimal places. Therefore, \(x<2.39\).

Key concepts

Inequality Calculation

When working with inequality calculations, the process closely resembles solving equalities, with a major difference being the direction of the inequality sign which affects how we interpret the solution set. In the given problem, we are asked to solve for a variable within an inequality, specifically \( \frac{1}{2.3x - 5.2} > 3.4 \). This type of problem requires a clear understanding of how to manipulate inequalities while maintaining their integrity.

To preserve the inequality, if we multiply or divide both sides of an inequality by a negative number, we must reverse the inequality sign. For instance, had the value \(2.3x - 5.2\) been negative, flipping the sign would have been necessary. This rule is pivotal and must be remembered at every step. In this problem, the multiplication by \(2.3x - 5.2\) does not necessitate flipping the inequality because we assume \(x\) is such that \(2.3x - 5.2 > 0\), in compliance with the original inequality.

The key steps to solve the inequality often involve removing fractions, distributing multiplication across terms, and finally isolating the variable to find its solution range. Each of these steps progresses toward a simplified form where the variable's solution range becomes evident.

Isolating Variables

To successfully isolate a variable in an inequality or equation, our goal is to manipulate the algebraic expression so that the variable stands alone on one side. This isolation process involves several algebraic operations such as addition, subtraction, multiplication, and division, applied to both sides of the inequality to maintain equality.

During the solution process for our inequality \( \frac{1}{2.3x - 5.2} > 3.4 \), the isolation begins with eliminating the fraction by multiplying both sides by \(2.3x - 5.2\), yielding \(1 > 3.4(2.3x - 5.2)\). By further distributing and then shifting constants and coefficients, we sequentially align the inequality in a form where \(x\) is by itself.

Remember the Balance

It's crucial to do to one side what we do to the other, preserving the balance of the inequality. In the example, adding \(17.68\) to both sides and then dividing by \(7.82\) maintains this balance, leading to \(x\) isolated on one side with its solution range on the other.

Rounding Decimal Places

The process of rounding decimal places is a numerical method used to reduce the decimal places in a number to a specified number of places, making the number easier to work with and understand. The exercise requests rounding to two decimal places. This means we want only two digits after the decimal point, with the second digit increased by one if the third digit (which will be discarded) is five or above.

In the context of our problem, after isolating \(x\) we obtained \(x < 2.3887218\). The third decimal place is \(8\), which is indeed more than five, therefore, we increase the second place \(8\) by one, leading to our rounded solution being \(x < 2.39\).

Significance in Inequalities

When dealing with inequalities, it's important to be cautious with rounding, as it can slightly alter the range of solutions that fit the inequality. To ensure accuracy within the context, always round as the last step, and follow the rounding rules carefully to avoid altering the solution's correctness.