Problem 60 - A Graphical Approach to Precalculus with Limits

Question

Use analytic or graphical methods to solve the inequality. $$2+\sqrt{3 x}<1$$

Step-by-Step Solution

Answer

There are no real solutions for the inequality.

Step 1: Isolate the square root

To isolate the square root, we start by subtracting 2 from each side of the inequality. This gives us \[ \sqrt{3x} < 1 - 2 \] Which simplifies to: \[ \sqrt{3x} < -1 \].

Step 2: Analyze the square root expression

The expression \( \sqrt{3x} \) is a square root, which implies it is always non-negative, meaning it is greater than or equal to zero for any real number \( x \). Hence, \( \sqrt{3x} < -1 \) has no real solutions, since a non-negative number can't be less than -1.

Key Concepts

Square Root Expression

Square root expressions involve taking the square root of a number or an expression. The square root, denoted by the radical symbol \( \sqrt{} \), essentially asks: "What number, when multiplied by itself, gives this number?"

When you see an expression like \( \sqrt{3x} \), it signifies finding the square root of whatever \( 3x \) evaluates to. Several key properties of square root expressions can help us in solving inequalities:

• Non-negativity: The square root of any real number is zero or positive. This is because no real number squared results in a negative number.


• Simplification: Use simplification rules, like factoring out squares, to make handling them more manageable.


• Behavior: Understanding the domain of the expression helps identify valid solutions. For example, under the square root sign, the expression needs to be non-negative.

Comprehending these properties simplifies the task and ensures that solutions are valid based on the square root's unique characteristics.

Analytic Methods

Analytic methods are a systematic approach to solving inequalities with mathematical principles and algebraic manipulation. These involve steps that align with properties of operations and expressions, and they facilitate understanding of underlying logic.

Here, we tackled the inequality \( 2 + \sqrt{3x} < 1 \) mathematically. The first action was to isolate the square root expression by manipulating all terms on one side of the inequality. By subtracting 2 from both sides, we got:

• \( \sqrt{3x} < -1 \)


• Recognizing that \( \sqrt{3x} \) and every square root expression is non-negative helps us conclude no real numbers make this inequality true.

Analytic methods aim to simplify and break down inequalities to judge feasibility of solutions, ensuring understanding of both approach and results.

Graphical Methods

Graphical methods involve drawing graphs to visualize equations or inequalities and find solutions visually. While this exercise utilized analytic approaches, graphical methods provide another invaluable way to interpret solutions.

In the context of the inequality \( \sqrt{3x} < -1 \), if attempted graphically:

• First, plot \( y = \sqrt{3x} \). This graph will show a curve starting from the origin (for the real number domain where \( x \geq 0 \)).


• Next, plot \( y = -1 \), which is a horizontal line below the x-axis.

The crucial observation here is that the curve representing \( \sqrt{3x} \) never descends below zero in the positive x-domain, reflecting the non-negative nature of square roots.

Therefore, visually, it’s clear that there are no x-values where \( \sqrt{3x} \) drops below -1, confirming what we've seen analytically. Using graphs complements understanding and offers another layer of verification between results from different methods.