Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Copy and complete the statement using the correct inequality symbol. If \(2 x \leq-8\), then \(x\) ________\(-4\).

Step-by-Step Solution

Answer

If \(2x \leq -8\), then \(x\) is less than or equal to \(-4\). (The correct inequality symbol to use is '\(\leq\)'.

Step 1 Divide both sides by 2

To isolate \(x\), divide both sides of the equation \(2x \leq -8\) by 2. That gives \(x \leq -4\).

Step 2: Identify the relevant trigonometric identities

Based on the given expression or equation, identify which trigonometric identities (Pythagorean, double-angle, sum/difference, etc.) are applicable.

Step 3: Apply the identities and simplify

Apply the identified identities to transform the expression. Simplify step by step, combining like terms and reducing fractions where possible.

Step 4: Solve or evaluate

If solving an equation, isolate the trigonometric function and find the angle(s). If evaluating, compute the final numerical value.

Step 5: State the result

Express the final answer, including all solutions in the required domain if solving an equation.

Step 6: Conclude with the answer

If \(2x \leq -8\), then \(x\) is less than or equal to \(-4\). (The correct inequality symbol to use is '\(\leq\)'.

Key Concepts

Solving Inequalities

When it comes to solving inequalities in algebra, understand that the process is quite similar to solving regular equations. However, instead of finding one exact number as an answer, you're determining a range of possible values that could work. To solve an inequality like \(2x \leq -8\), follow these straightforward steps:

Begin by isolating the variable on one side of the inequality. This requires the same operations you'd use for an 'equals' equation—addition, subtraction, multiplication, and division. In our example, the goal is to get \(x\) alone on one side. Therefore, you divide both sides of the inequality by 2. This operation yields \(x \leq -4\), which is your solution. It's important to note that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol, but this isn't necessary in this case because we're dividing by a positive 2.

Linear Inequalities

A linear inequality looks a lot like a linear equation, except it has an inequality symbol (\(>\), \(<\), \(\geq\), or \(\leq\)) instead of an equals sign. It is linear because the highest power of the variable is 1. The solution to a linear inequality is a range of values rather than a single value. For instance, if you're asked to solve \(2x \leq -8\), you're working with a linear inequality because the expression meets these criteria.

Upon solving, you ensure the \(x\) is alone on one side as you would with an equation. The solution \(x \leq -4\) indicates that any number less than or equal to -4 is a valid solution. This indicates a continuous range of answers, forming an interval. This is starkly different from solving a linear equation, which would just yield a single answer. To visually represent the solution, you could plot it on a number line, shading in all the numbers to the left of and including -4.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(x\) or \(y\)), and operations (such as addition and subtraction). Expressions are like sentences in the language of mathematics. What you need to remember is that expressions don't have an equals sign; that's what differentiates them from equations.

For example, \(2x - 3\) is an expression, while \(2x - 3 = 7\) is an equation. In the context of inequalities, the expression on one side of the inequality symbol is compared to the expression on the other side. In our initial problem \(2x \leq -8\), \(2x\) is the algebraic expression. While writing, calculating, or simplifying expressions, just focus on combining like terms and applying the order of operations correctly. They set the stage for understanding and solving mathematical statements, whether they're equations or inequalities.