Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, solve the inequality. Write your final answer in interval notation. $$ -\frac{1}{2} x \leq-\frac{5}{4}+\frac{2}{5} x $$

Step-by-Step Solution

Answer

The solution is \( \left[ \frac{25}{18}, \infty \right) \).

Step 1: Isolate x terms

First, move the \( \frac{2}{5}x \) from the right to the left side of the inequality to gather all the terms involving \( x \) on one side:\[-\frac{1}{2} x - \frac{2}{5} x \leq -\frac{5}{4}\]

Step 2: Combine like terms

Combine the \( x \) terms on the left side of the inequality. To do this, find a common denominator for \(-\frac{1}{2} x\) and \(-\frac{2}{5} x\) which is 10:\[-\frac{5}{10} x - \frac{4}{10} x \leq -\frac{5}{4}\] This simplifies to:\[-\frac{9}{10} x \leq -\frac{5}{4}\]

Step 3: Solve for x

To solve for \( x \), divide both sides by \(-\frac{9}{10}\). Remember that dividing by a negative number reverses the inequality sign:\[ x \geq \frac{-5/4}{-9/10} \] This simplifies to:\[ x \geq \frac{50}{36} \] which further simplifies to:\[ x \geq \frac{25}{18} \]

Step 4: Write in interval notation

The solution \( x \geq \frac{25}{18} \) means that \( x \) can be any number greater than or equal to \( \frac{25}{18} \). Therefore, in interval notation, this is written as:\[ \left[ \frac{25}{18}, \infty \right) \]

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of real numbers, often to represent solutions to inequalities. It provides a compact, efficient way to denote ranges of values that a variable might take. In interval notation, a common structure is used:

• Parentheses, \( ( ) \), indicate that an endpoint is not included (open interval).
• Brackets, \( [ ] \), indicate that an endpoint is included (closed interval).

For example, if a solution requires that a variable \( x \) is greater than or equal to \( \frac{25}{18} \), and doesn’t have an upper limit, you'll use interval notation to express it as \( \left[ \frac{25}{18}, \infty \right) \).
This means \( x \) can be any number starting from \( \frac{25}{18} \) all the way up to infinity. For inequalities, the range of solutions is easily comprehensible through this notation style, aiding in not just writing, but understanding mathematical inequalities.

Isolate x

Isolating \( x \) involves restructuring an equation or inequality such that \( x \) terms stand alone on one side. To start isolating \( x \) in the given inequality, the goal is to gather all \( x \) terms on one side.First, move any \( x \) terms to one side, while moving numerical constants to the opposite side.
In our inequality:\[-\frac{1}{2} x \leq -\frac{5}{4} + \frac{2}{5} x\]we moved \( \frac{2}{5}x \) to the left, meaning we subtract it from \( -\frac{1}{2}x \), forming:\[-\frac{1}{2} x - \frac{2}{5} x \leq -\frac{5}{4} \]
This step sets us up to "combine like terms" efficiently later, making it easier to solve for \( x \). Keeping \( x \) isolated simplifies the equation manipulation process and prepares a clear path towards solving the inequality.

Solve for x

Solving for \( x \) means finding the value or range of values that satisfies the equation or inequality. Once \( x \) is isolated, as shown before, the next steps become clearer.Upon combining like terms, the inequality turns into:\[-\frac{9}{10} x \leq -\frac{5}{4} \].
Here, \( x \) is further isolated by dividing both sides by \(-\frac{9}{10}\). Remember, dividing an inequality by a negative number reverses the inequality symbol:\[x \geq \frac{-5/4}{-9/10} \].
When simplified, this step results in:\[x \geq \frac{25}{18}\].
Thus, we found that the solution for \( x \) is any number greater than or equal to \( \frac{25}{18} \). Always pay attention to the change in the inequality sign when dividing by a negative; it's a common but crucial step in solving inequalities.

Combine Like Terms

Combining like terms is a simplification technique used to rearrange equations by grouping similar types of terms. Terms that include the same variables and powers should be added or subtracted together.Consider the case in our example:\[-\frac{1}{2} x - \frac{2}{5} x \leq -\frac{5}{4} \].
Both \(-\frac{1}{2}x\) and \(-\frac{2}{5}x\) are like terms because they involve \( x \). To combine them, transform each term to have a common denominator, making calculations straightforward.Using a common denominator of 10, the expression becomes:\[-\frac{5}{10} x - \frac{4}{10} x \leq -\frac{5}{4} \],which simplifies to:\[-\frac{9}{10} x \leq -\frac{5}{4} \].
This step reduces complexity, making it easier to identify the final solution when solving for \( x \). Efficiently combining like terms is a foundational practice leading to the correct simplification of broader mathematical problems.