Problem 47 - College Algebra

Question

Solve each linear inequality in Exercises 27-48 and graph the solution set on a number line. Express the solution set using interval notation. $$4(3 x-2)-3 x<3(1+3 x)-7$$

Step-by-Step Solution

Answer

The solution set to the inequality is all real numbers, or in interval notation \(-\infty, +\infty\), and every point on the number line will be shaded as it represents the solution set.

Step 1: Rearrange the inequality

To solve the inequality, the first step involves simplifying each side of the equation. This will result in \(12x - 8 -3x < 3 + 9x -7\). This simplifies to \(9x - 8 < 9x - 4\).

Step 2: Isolate x

Next, subtract \(9x\) from each side to leave \(x\) on one side. This gives \(-8 < -4\).

Step 3: Check the inequality result

This inequality statement \(-8 < -4\) is true because -8 is indeed less than -4. However, there is no \(x\) present in the final inequality. This means the inequality holds for every real number value of \(x\).

Step 4: Express in interval notation and graph

In interval notation, the solution set that includes all real numbers is represented as \(-\infty, +\infty\). The entirety of the number line represents the solution, so all points on the line would be shaded.

Key Concepts

Graphing Solution Sets on Number Lines

Visualizing inequalities on a number line is a beneficial skill when learning about linear inequalities. A number line provides a clear picture of all the possible values that satisfy an inequality. When graphing the solution set of an inequality, such as in our example where all real numbers are the solution, every point on the number line is included.

For inequalities that do not include all real numbers, open circles are used to represent values that are not included in the solution set (like with strict inequalities '<' and '>'), and closed circles are used for values that are included (like with inequalities '≤' and '≥'). Then you draw a line or ray to extend in the direction where all the numbers satisfy the inequality. This graphic representation helps students quickly understand the scope of solutions.

Interval Notation

Interval notation is a shorthand way to describe sets of numbers along a number line. In the context of linear inequalities, interval notation is efficient for conveying the range of solution sets. The solution to our example inequality is all real numbers, which we denote as \( -\infty, +\infty \).

Other common notations include using brackets [ or ] to indicate that a number is included in the set (for example, '[a, b]' means all numbers between and including a and b), and using parentheses ( or ) to indicate that a number is not included (for example, '(a, b)' means all numbers between a and b, not including a and b themselves). This symbolic representation is valuable because it compactly expresses which numbers satisfy an inequality.

Inequality Simplification

Simplifying inequalities is an essential step toward solving them. Just as you would with an equation, you combine like terms and simplify expressions on each side of the inequality before isolating the variable, if it is present. In our example, we began by expanding and simplifying terms to reduce the inequality to \(9x - 8 < 9x - 4\).

It's crucial to perform the same operations on both sides to maintain the balance of the inequality. Simplification might lead to a statement that no longer contains the variable, which indicates a specific kind of solution set—as was the case in our exercise, where the solution includes all real numbers because the resulting statement \( -8 < -4\) was always true.

Algebraic Manipulation

Algebraic manipulation involves using mathematical operations to rearrange and solve expressions and equations. In the case of inequalities, you follow the same principles as equations, but with an added layer of caution regarding the direction of the inequality. When multiplying or dividing by a negative number, the inequality sign flips direction.

For our exercise, algebraic manipulation was used to isolate the variable x by subtracting \(9x\) from both sides. This correct application of algebraic principles led to the discovery that the variable x was eliminated from the inequality, revealing that it was valid for all x in the real number set. Mastering algebraic manipulation is crucial for correctly solving and understanding linear inequalities.