Problem 15 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 7-x \geq 5 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, 2]\).

Step 1: Isolate the Variable

We start by isolating the variable on one side of the inequality. To do this, subtract 7 from both sides of the inequality: \[ 7 - x - 7 \geq 5 - 7 \]This simplifies to:\[-x \geq -2\]

Step 2: Solve for the Variable

Now, we solve for \(x\) by dividing both sides of the inequality by -1. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign:\[\frac{-x}{-1} \leq \frac{-2}{-1}\]This gives us:\[x \leq 2\]

Step 3: Express the Solution in Interval Notation

The solution \(x \leq 2\) can be expressed in interval notation by indicating that \(x\) can take any value from negative infinity up to and including 2. Thus, the interval notation is:\[ (-\infty, 2] \]

Step 4: Graph the Solution Set

To graph the solution set of \(x \leq 2\), we draw a number line and shade all values to the left of 2, including 2 itself. Use a closed dot on 2 to show that it is included in the solution.

Key Concepts

Solving Linear Inequalities

Linear inequalities, like equations, involve variables, constants, and operations such as addition, subtraction, multiplication, and division. However, rather than an equal sign, inequalities use symbols like \(<\), \(>\), \(\leq\), and \(\geq\). To solve a linear inequality:

• Isolate the variable on one side by performing appropriate operations on both sides of the inequality.
• Be cautious when multiplying or dividing by a negative number, as this action reverses the inequality sign. For instance, dividing both sides by -1 changes \(x \geq c\) to \(x \leq c\).

In our example, we started with \(7-x \geq 5\). By isolating \(x\), we ended with \(x \leq 2\). Always check your final inequality to ensure it logically follows from the steps taken.

Graphing Inequalities

Graphing inequalities on a number line helps visually represent the range of solutions. To graph a linear inequality:

• Draw a number line, placing key points (like numbers involved in the inequality) appropriately.
• Use an open dot for inequalities like \(<\) or \(>\), which do not include the endpoint, and a closed dot for \(\leq\) or \(\geq\), which do include the endpoint.
• Shade the portion of the number line that represents the solution set.

For the solution \(x \leq 2\), place a closed dot on 2 and shade everything to the left, indicating all numbers less than or equal to 2 are solutions. This visual cue helps confirm our algebraic solution.

Interval Notation

Interval notation is a streamlined method for expressing the set of solutions to inequalities. It uses parentheses \(()\) and brackets \([]\) to denote open and closed intervals, respectively. To convert an inequality to interval notation:

• Identify the direction of the inequality—from what value to what value does the solution set extend. For example, \(x \leq 2\) extends from negative infinity up to and including 2.
• Use \([\) if the number is included in the solution, and \(()\) if it isn't. Negative and positive infinity are always represented with \(()\) because they aren’t specific numbers, but rather concepts.

Thus, \(x \leq 2\) translates to \((-\infty, 2]\) in interval notation, showing all numbers less than or equal to 2 are part of the solution set.