Problem 60 - Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. \(2(-2) \leq 3 x-1\) and \(3 x-1 \leq-1-3\)

Step-by-Step Solution

Answer

Solution: \(x = -1\); Interval: \([-1, -1]\); Graph: Solid dot at -1.

Step 1: Simplify the Left Inequality

Let's begin by simplifying the first inequality. It is given as \(2(-2) \leq 3x - 1\). Simplify the left side:\[-4 \leq 3x - 1\]Next, to isolate \(3x\), add 1 to both sides:\[-4 + 1 \leq 3x\]This simplifies to:\[-3 \leq 3x\]

Step 2: Solve for x in Left Inequality

We continue by isolating \(x\) in the inequality \(-3 \leq 3x\). Divide every term by 3:\[\frac{-3}{3} \leq x\]This simplifies to:\[-1 \leq x\]

Step 3: Simplify the Right Inequality

Now, let's take on the second inequality \(3x - 1 \leq -1 - 3\). Start by simplifying the right side of the inequality:\[3x - 1 \leq -4\]Next, isolate \(3x\) by adding 1 to both sides:\[3x \leq -4 + 1\]This simplifies to:\[3x \leq -3\]

Step 4: Solve for x in Right Inequality

Next, solve for \(x\) in \(3x \leq -3\) by dividing every term by 3:\[x \leq \frac{-3}{3}\]This simplifies to:\[x \leq -1\]

Step 5: Combine the Inequalities

We have derived two inequalities: \(-1 \leq x\) and \(x \leq -1\). Combine them into a single compound inequality:\[-1 \leq x \leq -1\]

Step 6: Solution and Graph

The combined inequality \(-1 \leq x \leq -1\) indicates that \(x\) equals \(-1\). We represent this as a single point on the number line.Graph this solution by marking a solid dot at \(-1\). This dot indicates that \(-1\) is part of the solution. In interval notation, this is expressed as:\([-1, -1]\)

Key Concepts

Solving Inequalities

Solving inequalities involves finding the values of a variable that make the inequality true. In our compound inequality example, we have two separate inequalities that need to be solved together: \(-4 \leq 3x - 1\) and \(3x - 1 \leq -4\). The goal is to isolate the variable \(x\) on one side of each inequality.

• Step 1: Simplify each inequality by performing basic algebraic operations, such as addition or subtraction.
• Step 2: Use division or multiplication to solve for \(x\). Always remember to reverse the inequality sign if you multiply or divide by a negative number, but in our exercise, this rule wasn’t needed.

After simplifying, the inequalities were combined into a single compound inequality. This depicted the range of possible solutions that satisfy both inequalities.

Interval Notation

Interval notation provides a compact way of describing a range of numbers. It uses brackets to indicate whether endpoints are included in the interval. In the step-by-step solution, we solved for \(x\) which led to the compound inequality \(-1 \leq x \leq -1\). This indicates that the only value \(x\) can take is \(-1\).

In interval notation, this solution is written as \([-1, -1]\), meaning that the interval includes only the single point \(-1\).

• Use square brackets, \([\ ]\), when the endpoints are included in the solution set.
• Use parentheses, \((\ )\), if the endpoints are excluded from the solution.

Interval notation helps visualize solutions easily and is especially useful when dealing with complex ranges.

Graphing Solutions

Graphing solutions to inequalities helps in visualizing the range of values that make the inequality true. In the provided example, after solving the inequalities, we found that \(-1 \leq x \leq -1\), indicating that the only solution is when \(x = -1\).

• On a number line, you represent this by plotting a solid dot (or filled circle) at the point \(-1\). This shows that \(-1\) is included in the solution set.
• For compound inequalities with a range of solutions, you would shade the section of the number line between the endpoints to show all possible solutions.

The graph effectively communicates the solution set, allowing for quick comprehension of which values satisfy the inequality.