Problem 12 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ -4 x \geq 10 $$

Step-by-Step Solution

Answer

\(x \leq -\frac{5}{2}\); in interval notation: \((-\infty, -\frac{5}{2}]\); graph: closed dot at \(-\frac{5}{2}\) shading left.

Step 1: Isolate the Variable

To solve the inequality \(-4x \geq 10\), first isolate \(x\) by dividing both sides by \(-4\). Remember, when you divide or multiply both sides of an inequality by a negative number, you must reverse the inequality sign. So, the inequality becomes \(x \leq -\frac{10}{4}\) or \(x \leq -\frac{5}{2}\).

Step 2: Express the Solution in Interval Notation

The inequality \(x \leq -\frac{5}{2}\) can be expressed in interval notation. Since \(x\) includes all numbers less than or equal to \(-\frac{5}{2}\), the interval notation is \((-\infty, -\frac{5}{2}]\).

Step 3: Graph the Solution Set

To graph the solution set for \(x \leq -\frac{5}{2}\), draw a number line. Place a closed dot at \(-\frac{5}{2}\) to indicate that \(-\frac{5}{2}\) is included in the solution, then shade the line to the left towards \(-\infty\), indicating all numbers less than \(-\frac{5}{2}\) are part of the solution.

Key Concepts

Solving Inequalities

Inequalities are like equations, but they show a relationship where things are not equal. They use symbols like \(<\), \(>\), \(\leq\), or \(\geq\). Solving inequalities is similar to solving equations. The goal is to get the variable by itself on one side. However, there's an extra rule: if you multiply or divide by a negative number, flip the inequality sign.

For example, let's take \(-4x \geq 10\). We want to solve for \(x\). First, divide each side by \(-4\). But remember, dividing by a negative flips the sign. So, the inequality becomes \(x \leq -\frac{10}{4}\), or simplified, \(x \leq -\frac{5}{2}\).

This means any number \(x\) that's less than or equal to \(-\frac{5}{2}\) satisfies the original inequality. It's always good to double-check your steps to avoid mistakes, especially when dealing with negatives.

Interval Notation

Interval notation is a clever way to show all the solutions to an inequality. Instead of writing out every single number, you use a simple format. It looks like a pair of numbers inside parentheses or brackets. Here's how it works:

• Parentheses \(()\): These mean the endpoint is not included. Like \( (a, b) \).
• Brackets \([]\): These mean the endpoint is included. Like \( [a, b] \).
• Infinite symbols \(\infty\): When solutions extend indefinitely, use \(\infty\) or \(-\infty\). Never bracket infinities because they aren't exact numbers.

For the inequality \(x \leq -\frac{5}{2}\), \(-\frac{5}{2}\) is included in the solution, so we use a bracket here. The solution stretches infinitely to the left, so we use \(-\infty\). This gives us \(( -\infty, -\frac{5}{2} ]\) as the interval notation, representing all possible solutions.

Graphing Inequalities

Graphing inequalities is a handy visual tool that shows all solutions on a number line. It helps you see what numbers satisfy the inequality at a glance. Start by drawing a number line and mark the important numbers, then follow these steps:

• Closed Dot: If the inequality includes the number (like \(\leq\) or \(\geq\)), use a closed dot.
• Open Dot: If the inequality doesn’t include the number (like \(<\) or \(>\)), use an open dot.
• Shading: Shade the number line to show the direction of the inequality. Use an arrow or highlight.

For \(x \leq -\frac{5}{2}\), place a closed dot on \(-\frac{5}{2}\) and shade to the left, all the way towards \(-\infty\). This indicates that all numbers to the left are part of the solution. Graphs make understanding inequalities much clearer and more intuitive.