Question

Write an inequality that represents the statement and graph the inequality. \(x\) is less than \(-2\) and is at least \(-4\)

Short answer

The inequality that represents the statement is \(-4 \leq x < -2\). On the graph, there should be a closed circle at \(-4\) and an open circle at \(-2\), with a line running in between them.

Step-by-step solution

Interpret the Problem Statement

The problem gives us two conditions. One is that \(x\) is less than \(-2\), which can be written as \(x < -2\). The other condition is that \(x\) is at least \(-4\), that is \(x\) is greater than or equal to \(-4\), which can be written as \(x \geq -4\). Both of these conditions need to be true.

Represent the Conditions as an Inequality

The implications of the English 'and' is that both statements need to be true, therefore the two conditions can be combined into one inequality as such \(-4 \leq x < -2\). This inequality means that \(x\) is greater than or equal to \(-4\) and less than \(-2\), which adheres to the two conditions in the problem statement.

Graph the Inequality

On a number line, locate points at \(-4\) and \(-2\). Since \(x\) is allowed to be equal to \(-4\), the symbol at \(-4\) is a closed or filled circle. Meanwhile, since \(x\) isn't allowed to be \(-2\), the symbol at \(-2\) should be an open or empty circle. Draw a line between the two points to show that \(x\) can be any value between them.

Key concepts

Writing Inequalities

Writing inequalities is a foundational skill in algebra which involves translating phrases into mathematical statements. Begin by identifying key terms that suggest inequalities: phrases like 'less than', 'at least', or 'greater than' guide us. In our exercise, the phrase 'is less than (-2)' translates to the inequality symbol '<', resulting in the expression 'x < -2'. Similarly, 'is at least (-4)' implies that 'x' is on that value or higher, translating to 'x greater than or equal to -4', or 'x greater than or equal to -4' in mathematical terms. When these phrases are part of a compound statement using 'and', like in our exercise, each part is represented as part of a connected inequality.

Number Line Representation

Visualizing inequalities on a number line enhances understanding by providing a clear picture of all the possible solutions. To graph 'x greater than or equal to -4 and < -2', you begin by drawing a horizontal line and marking the critical values of -4 and -2. Given that 'x' can be equal to -4, a closed circle is placed at -4, symbolizing inclusion. Contrast this with an open circle at -2, indicating that 'x' does not reach this value. The area between -4 and -2 is then shaded or lined to show the range of acceptable values for 'x'. This visual representation on the number line is a powerful tool for understanding and communicating the solutions to inequalities.

Compound Inequalities

Compound inequalities are expressions with two inequality signs, describing a range of values that satisfy both inequalities. Think of it as the overlap between two sets. The compound inequality from our exercise, 'x greater than or equal to -4 and < -2', defines an intersection where 'x' must meet both criteria. This dual condition is achieved by combining the separate inequalities with the word 'and', which implies that only the numbers that satisfy both inequalities are solutions. In essence, compound inequalities capture a 'slice' or segment of the number line where all the conditions are true.

Solving Algebraic Inequalities

Solving algebraic inequalities involves finding all the values of a variable that make the inequality true. Our exercise is an example of this process where we interpret a written statement and translate it into a compound inequality. The solution to such problems requires understanding the rules for manipulation, much like equations, with close attention to changing the direction of the inequality when multiplying or dividing by a negative number. Solving inequalities is fundamental for real-world applications, such as determining ranges for measurements or calculating profits and expenses in business scenarios.