Problem 35 - Elementary Algebra

Question

Graph all solutions on a number line and give the corresponding interval notation. $$ x<-7 \text { or } x<2 $$

Step-by-Step Solution

Answer

The graph shows all values left of 2, in interval notation: \((-\infty, 2)\).

Step 1: Understand the Inequalities

We are given two inequalities: \( x < -7 \) and \( x < 2 \). These represent two conditions where \( x \) can be less than -7 or less than 2.

Step 2: Analyze Each Inequality

For the first inequality, \( x < -7 \), this means any value of \( x \) that is less than -7 satisfies this condition. For the second inequality, \( x < 2 \), any value of \( x \) less than 2 satisfies this condition.

Step 3: Graph Solutions on Number Line

- Draw a number line.- For \( x < -7 \), draw an open circle at -7 on the number line and shade all the values to the left.- For \( x < 2 \), draw an open circle at 2 on the number line and shade all the values to the left.- The fulfillment of the condition is anywhere below -7 or below 2, which effectively means below 2, as it encompasses below -7.

Step 4: Determine Interval Notation

Since the solutions cover all values less than 2, the interval notation is \((-\infty, 2)\). This interval means all real numbers to the left of 2 on the number line, not including 2 itself.

Key Concepts

Number Line

A number line is a straight, horizontal line that represents numbers in increasing order from left to right. It is a visual tool often used in mathematics to depict the position of numbers or to demonstrate inequalities graphically.

When working with inequalities, like in our given problem, the number line helps us show which parts of the line represent solutions to an equation. For instance, if you have an inequality such as \(x < -7\), you will represent it by marking an open circle on -7 and shading all values to the left, indicating all numbers less than -7 are solutions.

Another example is the inequality \(x < 2\), where you again mark an open circle, this time at 2, and shade leftward to show every number less than 2 satisfies the inequality. The open circle is essential because it visually denotes that the number itself (like -7 or 2) isn't included in the range of solutions.

Interval Notation

Interval notation is a concise way to describe a set of numbers between a start and end value. Instead of listing all numbers, it uses brackets to denote start and end points.

In the problem we analyzed, we are told that \(x < 2\). For interval notation, open parentheses \((\) are used to indicate numbers are not included in the interval range, i.e., values don't include the endpoint.

So, the numbers less than 2 are represented by the interval notation \((-finity, 2)\). The \(-\infty\) symbol indicates that the numbers continue indefinitely in the negative direction. Conversely, 2 is enclosed in a parenthesis to show that it is not included in the possible solutions. Thus, the interval notation effectively summarizes the entire solution set in a compact form.

Graphing

Graphing is a mathematical technique used to create graphs to visually represent solutions and different areas of equations, inequalities, and functions.

When graphing inequalities like \(x < -7\) and \(x < 2\), we plot a number line and use symbols such as open circles to indicate endpoints that are not part of the solutions. The regions that satisfy the inequalities will be shaded, illustrating all the valid values \(x\) can take.

This depiction helps us quickly see and understand where the solutions lie. It allows us to grasp, at a glance, the nature of the solutions and any overlaps or distinctions between different sets of values. Thus, graphing is not only an excellent tool for solving problems but also for communicating mathematical ideas clearly.