Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ 7 x-2 \leq 9 x+3 $$

Step-by-Step Solution

Answer

The solution is \([-2.5, \infty)\).

Step 1: Isolate Variables

First, we need to get all the terms involving the variable on one side of the inequality. Start by subtracting \(9x\) from both sides of the inequality: \(7x - 9x - 2 \leq 3\). This simplifies to \(-2x - 2 \leq 3\).

Step 2: Simplify the Equation

Next, simplify the left side of the inequality: \(-2x - 2 \leq 3\). To get rid of \(-2\), add 2 to both sides: \(-2x \leq 5\).

Step 3: Solve for x

Now, solve for \(x\) by dividing both sides of the inequality by \(-2\). Remember, dividing by a negative number reverses the inequality sign: \(x \geq -\frac{5}{2}\) or \(x \geq -2.5\).

Step 4: Express in Interval Notation

The solution \(x \geq -2.5\) can be written in interval notation as \([-2.5, \infty)\).

Step 5: Sketch the Solution on a Number Line

Draw a number line and make a solid dot at \(-2.5\) to indicate that it is included in the solution. Shade the line to the right of \(-2.5\) to show that all numbers greater than \(-2.5\) are included.

Key Concepts

Interval Notation

Interval notation is a way of representing a range of numbers that satisfy an inequality. It provides a simple and clear way to describe the solution set. You might see it written as \[ [a, b] \]or \[ (a, b) \].Here's what these symbols mean:

• Square brackets like \([a, b]\) indicate that the endpoints \(a\) and \(b\) are included in the interval (closed interval).
• Round brackets like \((a, b)\) indicate the endpoints are not included in the interval (open interval).
• An interval like \( [-2.5, \infty) \) means the solution set starts from -2.5 and goes towards positive infinity. The square bracket at \(-2.5\) means -2.5 is included, and the round bracket at infinity indicates it's not a specific limit, as infinity is limitless.

By using interval notation, you can easily express the solution to inequalities in a brief format. Try to visualize the number line as this makes understanding this concept much easier!

Number Line

A number line is a visual representation of numbers plotted in a straight line. It's a powerful tool to help you understand and solve inequalities. Here's how you can use a number line:

• Mark a point on the number line corresponding to the number in the inequality.
• If the inequality includes the number (like \( x \geq -2.5 \)), make a solid dot at that point. A solid dot means that the point is part of the solution set.
• If the number is not included (for example, \( x > -2.5 \)), use an open circle instead. An open circle indicates that the point is not part of the solution set.
• Draw an arrow or shade the number line in the direction where the inequality holds. For example, for \( x \geq -2.5 \), you would shade to the right of -2.5, covering all numbers greater than or equal to -2.5.

Using a number line helps you better grasp the idea of what solutions to inequalities look like visually, making it easier to understand.

Reverse Inequality

When solving inequalities, sometimes you must divide or multiply by a negative number. This is when the concept of reversing inequalities becomes crucial. Here's why:

• Whenever you multiply or divide both sides of an inequality by a negative number, you must switch the inequality sign.
• This swap is necessary because the order of numbers changes when you multiply by negative values. For example, if \(-2 < 3\)is true, multiplying both sides by -1 gives \(2 > -3\),changing the inequality sign from \(<\) to \(>\).
• In the exercise above, we ended with \(-2x \leq 5\),and dividing both sides by \(-2\), changed the inequality to \(x \geq -2.5\). That flip of the inequality direction is key when handling negative coefficients!

Understanding this rule is a cornerstone in solving inequalities accurately.

Solving Inequalities

Solving inequalities is somewhat similar to solving equations, but with a few extra rules. Here’s a step-by-step approach:

• First, get all the variable terms on one side and numbers on the other by adding or subtracting appropriate terms from both sides. For example, in the given problem: \(7x - 9x - 2 \leq 3\).
• Simplify both sides as much as possible. Maintain balance by performing the same operation on both sides.
• Be careful when multiplying or dividing by negative numbers. Remember this rule: the inequality sign should be reversed.
• Once isolated, interpret your result. For instance, \(x \geq -2.5\)means that all values of \(x\) equal to or greater than -2.5 satisfy the inequality.
• Draw the result on a number line and express your solution in interval notation for clarity.

While it may seem complex at first, with practice, you'll be able to solve inequalities swiftly and with confidence!