Problem 28 - Algebra and Trigonometry

Question

In all exercises other than \(\varnothing\), use interval notation to express solution sets and graph each solution set on a number line. In Exercises \(27-50,\) solve each linear inequality. $$ 2 x+5<17 $$

Step-by-Step Solution

Answer

The solution to the inequality \(2x + 5 < 17\) is \(x < 6\), which is expressed in interval notation as \(-\infty, 6)\) and graphed on a number line with an open circle at 6 and an arrow pointing towards negative infinity.

Step 1: Simplify the Linear Inequality

Start by subtracting 5 from both sides of the inequality \(2x + 5 < 17\): \(2x + 5 - 5 < 17 - 5\), simplifies to \(2x < 12\).

Step 2: Solve for the Variable x

Continue by dividing each side of the inequality by 2: \( \frac{2x}{2} < \frac{12}{2}\), leaves \(x < 6\). This is the solution for the inequality.

Step 3: Express the Solution Set in Interval Notation

The solution \(x < 6\) in interval notation is written as \(-\infty, 6)\). The parenthesis means that 6 is not included in the solutions.

Step 4: Graph the Solution Set on a Number Line

The number line needs asymptote pointing towards \(-\infty\) from 6, an open circle at 6 indicating that 6 is not included in the solution set.

Key Concepts

Interval Notation

Interval notation offers a concise way to represent ranges of numbers on the real number line. In the context of inequalities, this form of notation is particularly useful for visually summarizing the solution set.
For example, consider the linear inequality solution, which states that for all values of \(x \) that satisfy the inequality \(2x + 5 < 17\), \(x\) must be less than 6. To express this range of values in interval notation, we write it as \((-\infty, 6)\). This means that \(x\) can be any number less than 6, but not 6 itself, as it's not included in the solution set. The symbol \(-\infty\) indicates that there's no lower bound to the range, while the opening parenthesis \(()\) indicates that the upper bound, 6 in this case, is not part of the interval.
In contrast, a closed interval, such as \([3, 8]\), would imply that both endpoints—3 and 8—are included in the set. The square brackets \([]\) denote that the endpoints are part of the interval. Understanding interval notation is crucial, as it allows for clear and efficient communication of solution sets in mathematics.

Number Line Graphing

Graphing an inequality on a number line is a visual means of representing the solution set. For the inequality \(x < 6\), derived from the given problem \(2x + 5 < 17\), one would create a number line with a mark at the number 6. An open circle is then placed at 6 to indicate that this value is not included in the solution—it's an open interval. An arrow extending to the left from the open circle towards negative infinity (\(-\infty\)) visually communicates that all numbers less than 6 are included in the solution set.
This approach to graphing helps to visually reinforce the concept of which values are included in the solution set and whether these values approach a certain limit from the left (as with negative values) or from the right (as with positive values). The simplicity of number line graphs makes them an invaluable tool in the study of inequalities, particularly for those who are more visually-oriented learners.

Linear Inequality Simplification

Simplifying a linear inequality involves isolating the variable on one side of the inequality to find its possible values. In the provided step by step solution, this simplification process begins by eliminating the constant term from one side, as seen when the term \(+5\) is subtracted from both sides of the inequality \(2x + 5 < 17\), yielding \(2x < 12\).

• Following this, we need to isolate the variable \(x\). As \(x\) is initially multiplied by 2, we must perform the inverse operation, division, to free \(x\) from this coefficient. Dividing each side of the inequality \(2x < 12\) by 2 gives us \(x < 6\).
• This step not only simplifies the inequality but also clearly reveals the set of possible values for \(x\) that satisfy the original inequality.
• Simplification is vital as it makes it easier to understand and work with inequalities, whether we're graphing them, using them to predict values, or incorporating them into more complex mathematical problems.

By mastering inequality simplification, students can efficiently solve a broad spectrum of problems, a skill that extends well beyond homework exercises into real-world applications.