Problem 61 - Algebra 1

Question

Complete the statement using \(>,<, \geq,\) or \(\leq\). If \(-7 \geq w\) then \(w\)__ -7

Step-by-Step Solution

Answer

If \(-7 \geq w\) then \(w \leq -7\).

Step 1: Understanding the Inequality

The inequality \(-7 \geq w\) meaning that the value of w is less than or equal to -7. Hence, when we compare \(w\) to -7, it should be recognized as less than or equal to -7.

Step 2: Writing the Solution

Thus, the proper way to complete the statement is: If \(-7 \geq w\) then \(w \leq -7\). This says that if -7 is greater than or equal to w, then w is less than or equal to -7, which is the correct interpretation.

Key Concepts

Solving Inequalities

Understanding how to solve inequalities is a fundamental skill in algebra that involves finding the values of a variable that make an inequality true. An inequality is like an equation, but instead of an equal sign (\text{=}), it involves symbols like greater than (\text{>}), less than (\text{<}), greater than or equal to (\text{\geq}), or less than or equal to (\text{\leq}).

To solve an inequality, you follow similar steps as with equations, such as adding or subtracting the same value from both sides or dividing both sides by a positive number. Be mindful, though, that multiplying or dividing both sides by a negative number reverses the inequality symbol. For example, if you multiply both sides of the inequality \text{\(5 > -3\)} by \text{-1}, you get \text{-5 < 3}.

When presenting the solution, it is critical to display the answer in a way that clearly shows the range of values that solve the inequality. This is often expressed with a number line or by writing the answer in interval notation, which is compact and precise.

Inequality Notation

Inequality notation is a system of symbols and formatting used to express the relationship between values in which they are not necessarily equal, but greater or lesser in value. It's essential to know and effectively use these symbols to interpret and solve inequalities correctly.

Common inequality symbols include:

• Greater than (\text{>}): This signifies that the value on the left is larger than the one on the right.
• Less than (\text{<}): This denotes that the value on the left is smaller than the one on the right.
• Greater than or equal to (\text{\geq}): Indicates that the left value is either larger than or exactly equal to the right value.
• Less than or equal to (\text{\leq}): Signals that the left value is either less than or exactly equal to the right value.

To write the solution to an inequality, use the correct symbol to describe the relationship. For instance, if we determine that a value \text{\(x\)} is greater than 3, we write \text{\(x > 3\)}. Conversely, if \text{\(x\)} must be less or equal than 3, we write \text{\(x \leq 3\)}. Understanding these notations helps in interpreting and solving algebraic inequalities.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables (such as \text{\(x, y, w\)}), and arithmetic operations (like addition, subtraction, multiplication, and division). They provide a way to generalize problems and work with unknown values. It's essential to understand how to manipulate these expressions using algebraic rules.

An expression can take many forms. For simple ones, like \text{\(2x + 3\)}, we can evaluate them for specific values of \text{\(x\)}. For more complex expressions that include inequalities, like \text{\(x > y + 2\)}, we interpret this as saying that \text{\(x\)} must be greater than whatever value comes from adding 2 to \text{\(y\)}.

The manipulation of algebraic expressions lies at the heart of solving algebraic inequalities. By adding or subtracting terms, multiplying or dividing by constants, and factoring, we can transform complex inequalities into simpler forms, revealing the values that satisfy the inequality. Always check your final solution by substituting back into the original inequality to ensure that it makes the inequality true.