Question

Copy and complete the statement using the correct inequality symbol. If \(3 x \leq-15\), then \(x\) ______\(-5\).

Short answer

The correct inequality symbol to complete the statement is '\( \leq \)'.

Step-by-step solution

Solving the given inequality

Start by solving the given inequality for \(x\). To do this, both sides of the inequality \(3x \leq -15\) should be divided by 3. That leads to \(x \leq -5\).

Substitute and fill in the gap

Now, substitute this answer into the blank in the given sentence. After substituting the value, the statement will be: If \(3x \leq -15\), then \(x \leq -5\).

Key concepts

Solving Inequalities

Solving inequalities is a fundamental skill in algebra that helps us find a range of possible solutions rather than a single value. When faced with an inequality like \(3x \leq -15\), our task is to determine the values of \(x\) that make this statement true. Here's a straightforward way to approach this:

• Start by treating the inequality as if it were an equation. This means performing the same operations on both sides to maintain balance.
• In our example, divide both sides of \(3x \leq -15\) by 3 to isolate the variable \(x\). This results in \(x \leq -5\).

Remember the rules when solving inequalities. If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign. In this example, we divided by a positive number, so the sign stays the same.

Inequality Symbols

Inequality symbols are crucial as they show the relationship between two expressions. Let's explore some common symbols:

• \(<\) and \(>\) mean "less than" and "greater than" respectively.
• \(\leq\) and \(\geq\) mean "less than or equal to" and "greater than or equal to".

In our exercise, we used \(\leq\), indicating that \(x\) can be any value less than or equal to -5. It's essential to understand these symbols, as they define the set of solutions we get after solving the inequality. Whenever you encounter these symbols, consider both numerical and graphical methods to visualize the solutions better.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent a specific value. They are the building blocks of equations and inequalities.
In our example, the expression \(3x\) represents a value that depends on \(x\). When solving \(3x \leq -15\), you manipulate the expression to find what \(x\) must be for the inequality to hold true.
Understanding algebraic expressions involves:

• Recognizing variables and constants and knowing how they interact through operations like addition and multiplication.
• Using properties of operations to simplify expressions, such as combining like terms or factoring.

This understanding allows you to not only solve inequalities but also tackle more complex algebraic problems with confidence.