Problem 47 - College Algebra Essentials

Question

Use interval notation to express solution sets and graph each solution set on a number line. Solve each linear inequality. $$4(3 x-2)-3 x<3(1+3 x)-7$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x\) belongs to \((-\infty, +\infty)\)

Step 1: Distribute the Multiplication Inside Parentheses

Simplify both sides of the inequality using the distributive property. This gives us \(12x - 8 - 3x < 3 + 9x - 7\). Then further simplify to \(9x - 8 < 9x - 4\).

Step 2: Regroup the Terms

After simplifying, rearrange by keeping variable 'x' on one side and the constants on the other side. This gives us \(9x - 9x < -4 + 8\).

Step 3: Solve for 'x'

Now, simplify the inequality to get the solution for 'x'. This results in \(0 < 4\). As '0' is less than '4', this holds true for all values of 'x'. Therefore, the solution set for 'x' is \(-\infty < x < +\infty\).

Step 4: Graph the solution

On the number line, the solution \(-\infty < x < +\infty\) represents all numbers, meaning every point on the line is a solution. So, you will shade the entire number line.

Key Concepts

Interval Notation

Interval notation is a method used to describe a set of numbers between two endpoints. It's commonly used in mathematics to express the solution sets of inequalities. Understanding interval notation is akin to knowing the shorthand of mathematical communication, allowing us to succinctly represent ranges of numbers. In the context of linear inequalities, interval notation provides a clear and compact way to present the values that satisfy the inequality.

For instance, when we say the solution to an inequality is \(0 \< x \< 4\), in interval notation, we write this as \(0, 4\). Open intervals, denoted by parentheses \( \), indicate that the end values are not included in the solution. Conversely, closed intervals, denoted by brackets \[ \], include the endpoints as part of the solution. The solution \( -\infty \< x \< +\infty \) in interval notation is expressed as \(-\infty, +\infty\), which signifies that all real numbers are included in the solution set, emphasizing the idea that 'x' can be any number.

Distributive Property

The distributive property is a fundamental principle in algebra that allows us to multiply a single term by each term inside a parenthesis. It aids in the simplification of algebraic expressions and is crucial when solving linear inequalities like our given exercise.

When applied, we multiply the outside value by each term within the parentheses. For example, in the expression \(4(3x - 2)\), according to the distributive property, we multiply 4 by each term inside the bracket, yielding \(12x - 8\). This property is also reversible, which means you can factor out common terms from an expression. Simplifying both sides of an inequality using the distributive property helps isolate the variable and makes it easier to find the solution. This step is crucial for understanding inequalities as it transforms them into a more manageable form that can be further simplified and solved.

Number Line Graphing

Number line graphing is a visual representation of numbers on a straight line where each point on the line corresponds to a number. In algebra, it's an indispensable tool for visualizing the solution sets of inequalities. Graphing solutions on a number line provides an intuitive way to see the range of possible values that satisfy an inequality.

For example, the solution to an inequality expressed as \(0 \< x \< 4\) is graphed by marking the numbers '0' and '4' on the number line and shading the segment in between, but without including the endpoints, to indicate that '0' and '4' are not part of the solution. In the case where the solution is all real numbers, as in our exercise where \( -\infty \< x \< +\infty\), the entire number line is shaded to represent that there's no restriction on the value of 'x'. Thus, number line graphing serves as an easy-to-read map to the solution of inequalities.