Problem 73 - College Algebra with Modeling and Visualization

Question

Solve each inequality numerically. Write the solution set in set-builder or interval notation, and approximate endpoints to the nearest tenth when appropriate. $$ 1 \leq 3 x-2 \leq 10 $$

Step-by-Step Solution

Answer

The solution set is \([1, 4]\).

Step 1: Write the Double Inequality as Two Separate Inequalities

The given inequality is: \[1 \leq 3x - 2 \leq 10\]This can be split into two separate inequalities:1. \(1 \leq 3x - 2\)2. \(3x - 2 \leq 10\)

Step 2: Solve the First Inequality

We will solve the inequality \(1 \leq 3x - 2\) by adding 2 to both sides to simplify it:\[1 + 2 \leq 3x\]\[3 \leq 3x\]Divide both sides by 3 to solve for \(x\):\[\frac{3}{3} \leq x\]\[1 \leq x\]

Step 3: Solve the Second Inequality

Next, we solve the inequality \(3x - 2 \leq 10\). Add 2 to both sides:\[3x - 2 + 2 \leq 10 + 2\]\[3x \leq 12\]Divide both sides by 3:\[x \leq \frac{12}{3}\]\[x \leq 4\]

Step 4: Combine the Two Inequalities

The solution to the double inequality combines the solutions of the two separate inequalities:From Step 2, we have \(1 \leq x\).From Step 3, we have \(x \leq 4\).Thus, the combined solution is \(1 \leq x \leq 4\).

Step 5: Write the Solution in Interval Notation

The solution \(1 \leq x \leq 4\) can be written in interval notation as:\[[1, 4]\]

Key Concepts

Double Inequalities

A double inequality is a way of expressing two related inequalities in one statement. For example, the original inequality \(1 \leq 3x - 2 \leq 10\) is a double inequality. It tells us that \(3x - 2\) is greater than or equal to 1 and at the same time less than or equal to 10.

To solve a double inequality, we break it into two separate inequalities:

• \(1 \leq 3x - 2\)
• \(3x - 2 \leq 10\)

The benefit of this is it allows us to handle each inequality as an individual problem. We solve each of these inequalities independently and then combine the solutions to find the full answer to the original double inequality. When solving these inequalities, remember to treat both sides independently through similar operations, but ensure you maintain the inequality throughout.

Interval Notation

Interval notation is a succinct way of expressing a range of values that a variable may take. It's especially useful in displaying solutions to inequalities. For example, once we determine that the solution set for the inequality \(1 \leq x \leq 4\) is \(x\) taking values between 1 and 4, we can easily express this in interval notation as \([1, 4]\).

In interval notation, the brackets \([] and []\) or parentheses \(() and ())\) indicate whether endpoints are included or not:

• \([\) signifies the endpoint is included, called "closed interval."
• \(()\) signifies the endpoint is not included, called "open interval."

For this problem, since both 1 and 4 are included in the solution set, we use square brackets: \([1, 4]\).

Interval notation thus provides a quick and clear method to represent solutions to inequalities, with arguably more visual clarity than writing it out in words or set-builder notation.

Algebraic Solution

The algebraic solution involves manipulating the original inequality using algebraic operations to isolate the variable and calculate its possible values. Here are the steps to achieve that for the inequality \(1 \leq 3x - 2 \leq 10\):

• Step 1: Add 2 to both parts of the inequality to eliminate the -2, resulting in: \(3 \leq 3x \leq 12\).
• Step 2: Divide each part of the inequality by 3, the coefficient of \(x\), which simplifies it to \(1 \leq x \leq 4\).

This shows us explicitly how \(x\), the variable, ranges between 1 and 4 inclusive. It's crucial when performing these operations to maintain the direction of our inequality, confirming the solution remains valid through each transformation step.

Numerical Approximation

Numerical approximation is often applied to find more precise or applicable solutions to inequalities when exact values are not easily determined or when working with decimals. In our example, each value of \(x\) in the range between 1 and 4 already gives a precise solution, thus requiring no further approximation. However, should you have endpoints that don't divide evenly, numerical approximation can be invaluable.

Here's when you might use it:

• If solving gives a result like \(x \leq 3.9999\), we can approximate to \(x \leq 4\) when decimals are unnecessary.
• In more complex scenarios, you might need to round endpoints to the nearest thousandth or tenth, depending on the requirement given in a specific context.

Make sure you know whether an approximate solution is sufficient or needed and be comfortable switching between exact and approximate solutions. This flexibility allows you to apply mathematics effectively in a wide range of real-world situations.