Problem 5 Express the given inequality in ... [FREE SOLUTION]

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Intermediate Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 2: Problem 5

Express the given inequality in interval notation and sketch a graph of the interval. $x<-2$

Short Answer

Expert verified

The interval is $(-\infty, -2)$, shown with an open circle at $-2$ and shaded to the left.

Step by step solution

Understand the Inequality

The given inequality is $x < -2$. This means that $x$ includes all numbers that are less than $-2$. The inequality is a strict inequality, indicating that $-2$ itself is not included.

Convert to Interval Notation

Since $x$ includes all numbers less than $-2$ but not $-2$ itself, we express this using interval notation as $(-\infty, -2)$. Intervals with infinity are always open intervals and the interval is opened on negative 2 as well, since negative 2 is not included.

Sketch the Graph

To graph the interval $(-\infty, -2)$, draw a number line. Place an open circle at $-2$ to indicate that $-2$ is not included in the interval. Then shade the line to the left of $-2$ to represent all numbers less than $-2$. The open circle signals the end point is not part of the interval.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inequality Graphing

Graphing an inequality like $x < -2$ helps visualize the solution set on a number line. To represent this inequality correctly:

First, identify the critical value where the change occurs, which is $-2$ in this case.
Since it鈥檚 a strict inequality $(<)$, $-2$ is not part of the solution, so we use an open circle to clearly highlight this.
Next, shade the line extending to the left of $-2$ to indicate all values less than $-2$ satisfy the inequality.

Graphing inequalities aids in understanding which numbers satisfy a condition, providing a visual tool alongside the mathematical representation.

Number Line

A number line is a visual representation aiding in the clear understanding of numbers and inequalities. When sketching a number line for inequalities:

Draw a horizontal line with evenly spaced markings representing different numbers.
Identify the crucial points relevant to your inequality - here it's $-2$.
An open circle is placed on the number line at critical points not included in the solution, such as at $-2$ for $x < -2$.
Shade the area on the line that represents the solution. For this < inequality, shade to the left of the open circle.

Number lines provide a straightforward method to see solutions of inequalities and ensure you correctly interpret the math.

Strict Inequalities

Strict inequalities, such as $x < -2$, are a type of inequality where the number on the boundary is not part of the solution set. Unlike non-strict or inclusive inequalities ($\leq$ or $\geq$), strict inequalities:

Do not include the boundary number, meaning for $x < -2$, $-2$ isn鈥檛 included.
Represented graphically, the non-inclusive nature is shown with an open circle.
Interval notation for strict inequalities always uses parentheses, not brackets, as is the case with < $(-< \infty, -2)$.

Understanding strict inequalities is crucial for correctly interpreting and communicating mathematical concepts in both written and graphical forms.

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91影视

Express the given inequality in interval notation and sketch a graph of the interval. \(x<-2\)

Short Answer

Step by step solution

Understand the Inequality

Convert to Interval Notation

Sketch the Graph

Key Concepts

Inequality Graphing

Number Line

Strict Inequalities

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