Question

Copy and complete the statement using the correct inequality symbol. If \(2 x>6\), then \(x \longrightarrow 3\)

Short answer

In the statement \(x \longrightarrow 3\), the correct inequality symbol is '>', so the completed statement is \(x > 3\).

Step-by-step solution

Analyze the given inequality

The given inequality is \(2x > 6\). In order to isolate \(x\), you need to divide both sides of the inequality by 2.

Isolate x

Dividing both sides of the inequality \(2x > 6\) by 2, you get \(x > 3\).

Comparing x with the statement

Comparing \(x > 3\) with the statement \(x \longrightarrow 3\), the correct inequality symbol is '>'.

Key concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols; it is a language that allows us to describe patterns, relationships, and changes.

In our problem, we are given the algebraic expression \(2x > 6\). Here, \(2x\) represents twice the value of \(x\), and the '>' symbol represents an inequality – specifically, that the quantity on the left is greater than the quantity on the right. Algebra helps us to manipulate this expression in a way that we can find the value of \(x\) that satisfies the inequality.

Understanding algebra is crucial as it enables us to solve equations and inequalities by performing operations such as addition, subtraction, multiplication, and division to both sides of an equation or an inequality, thereby maintaining the balance and the truth of the original statement.

Inequality Symbols

Inequality symbols are part of the foundational vocabulary in algebra. They are used to express the relationship between two values. The most commonly used symbols are '<' (less than), '>' (greater than), '\(\leq\)' (less than or equal to), and '\(\geq\)' (greater than or equal to).

With the example \(2x > 6\), the '>' symbol indicates that the value of \(2x\) is greater than 6. If the initial inequality were \(2x < 6\), it would mean that the value of \(2x\) is less than 6. These symbols are crucial in mathematics because they help us understand and communicate the precise relationship between numbers or algebraic expressions.

When solving these inequalities, the direction of the inequality symbol will only change if we multiply or divide both sides by a negative number. Otherwise, the direction remains the same as we perform our operations to isolate the variable.

Isolating Variables

Isolating variables is a fundamental technique in algebra that involves rearranging an equation or inequality so that the variable we are solving for is on one side of the equation by itself.

In our given problem, we need to isolate \(x\) in the inequality \(2x > 6\). The process involves performing the same mathematical operation to both sides of the inequality in order to solve for \(x\). Since \(x\) is being multiplied by 2, we reverse this operation by dividing both sides by 2, maintaining the balance and preserving the inequality relationship. This results in \(x > 3\), which means any number greater than 3 will satisfy the original inequality.

Understanding how to isolate variables is critical to solving equations and inequalities in algebra. By mastering this skill, one can efficiently find the values that satisfy the given equation or inequality, which is an essential step in the mathematical problem-solving process.