Problem 59 - Elementary and Intermediate Algebra

Question

(surrounding them by spaces should be sufficient). Solve each inequality. Write the solution set in interval notation and then graph it. $$ -3 x-1 \leq 5 $$

Step-by-Step Solution

Answer

The solution is \([-2, \infty)\) and it's graphed with a closed dot on -2 and an arrow to the right.

Step 1: Isolate the variable term

Start with the inequality:\[-3x - 1 \leq 5\]Add 1 to both sides to begin isolating the term with \(x\).\[-3x - 1 + 1 \leq 5 + 1\]Simplify both sides:\[-3x \leq 6\]

Step 2: Solve for the variable

Now, divide both sides by -3. Remember, when dividing or multiplying both sides of an inequality by a negative number, the inequality sign flips.\[x \geq \frac{6}{-3}\]Simplify:\[x \geq -2\]

Step 3: Write the solution in interval notation

Convert the inequality \(x \geq -2\) into interval notation. Since \(x\) includes -2 and every larger number:\[[-2, \infty)\]

Step 4: Graph the solution

Draw a number line. Place a closed dot at -2, showing that -2 is included in the solution. Draw an arrow to the right, indicating all numbers greater than -2 are included. This represents the interval \([-2, \infty)\).

Key Concepts

Interval Notation

Interval notation is a way to describe sets of numbers along a number line, using specific symbols to indicate whether the endpoints are included or not. For instance, brackets "\([ ]\)" indicate that a number is included in the interval, while parentheses "\(( )\)" suggest it is not.

In the inequality solution from our exercise, you finished with the inequality \(x \geq -2\). This means that \(x\) can be any number starting from \(-2\) and moving toward infinity. To express this as interval notation, we write \([-2, \infty)\). Note the closed bracket at \(-2\) because \(-2\) is part of the solution, while infinity is always accompanied by a parenthesis as it is not a fixed number.

Understanding interval notation helps you clearly and precisely convey solution sets for inequalities.

Inequality Solution

Solving inequalities is similar to solving equations but with a few extra rules. The main concept is to isolate the variable on one side of the inequality sign \(\leq, \geq, <, \) or \( > \).

1. Begin by simplifying both sides, if needed, to isolate terms.2. When you multiply or divide both sides of an inequality by a negative number, remember this crucial rule: flip the inequality sign. So, \(\leq\) becomes \(\geq\), and \(<\) becomes \(>\), and vice versa.

In our problem, starting from \(-3x - 1 \leq 5\), you first added 1 to clear the constant term, resulting in \(-3x \leq 6\). Dividing by \(-3\) both simplifies \(x\) and flips the inequality, resulting in \(x \geq -2\).

Always perform a check by plugging values back into the original inequality to ensure accuracy.

Graphing Inequalities

Graphing inequalities visually represents all possible solutions on a number line. This approach helps you see, at a glance, which values satisfy the inequality.

To graph \(x \geq -2\):

• First, locate \(-2\) on the number line.
• Place a closed dot at \(-2\) because the inequality sign "\(\geq\)" means \(-2\) is included in the solution.
• Draw an arrow extending to the right from \(-2\) to indicate all numbers greater than \(-2\) are included.

This representation matches the interval notation \([-2, \infty)\). By doing this, you communicate the comprehensive set of values that satisfy the original inequality.