Problem 53 - Elementary and Intermediate Algebra

Question

Solve each compound inequality, if possible. Graph the solution set (if one exists) and write it using interval notation. $$ 5(x-2) \geq 0 \text { and }-3 x<9 $$

Step-by-Step Solution

Answer

The solution set is \([2, \infty)\).

Step 1: Solve the first inequality

For the first inequality, we'll solve \(5(x-2) \geq 0\). Begin by distributing the 5: \(5x - 10 \geq 0\). Next, add 10 to both sides to isolate the term with \(x\): \(5x \geq 10\). Finally, divide both sides by 5 to solve for \(x\): \(x \geq 2\).

Step 2: Solve the second inequality

Now, solve the second inequality \(-3x < 9\). Start by dividing both sides by -3, remembering to reverse the inequality sign: \(x > -3\).

Step 3: Find the intersection of solutions

The solution to the compound inequality 'and' is the intersection of the two solutions found. Combine \(x \geq 2\) and \(x > -3\) to get the region where both conditions are satisfied: \(x \geq 2\). This is because \(x > -3\) imposes no additional restriction in the presence of \(x \geq 2\).

Step 4: Represent the solution in interval notation

The solution \(x \geq 2\) can be expressed in interval notation as \([2, \infty)\).

Step 5: Graph the solution set

On a number line, draw a closed dot or a solid point at 2 and shade all numbers to the right, extending to infinity. This indicates all numbers greater than or equal to 2 are included in the solution set.

Key Concepts

Graphing Inequalities

Graphing inequalities on a number line helps visualize the solution. For our compound inequality, we solved it and found that the solution is given by \(x \geq 2\), meaning \(x\) can be any number greater than or equal to 2. To represent this on a number line:

• Draw a solid dot at 2. This shows that 2 is included in the solution set, as indicated by the \(\geq\) sign.
• Shade the entire region to the right of 2. This illustrates that the solution includes all numbers greater than 2, extending infinitely to the right.

In compound inequalities that involve "and," like ours, we look for the overlap of solutions from individual inequalities. Here, since one solution was \(x \geq 2\) and the other was \(x > -3\), the stricter condition \(x \geq 2\) completely satisfies the requirement, thus dictating the overall graphing.

Interval Notation

Interval notation provides a concise way of expressing a set of numbers on the number line. Consider it a shorthand that eliminates the need for lengthy explanations.
To write \(x \geq 2\) in interval notation:

• Use a square bracket \([\) when the boundary number is included in the set, like in 2.
• Use a parenthesis \()\) when the set continues beyond without including a boundary number, such as \(\infty\).

Combine these practices for our example, which results in \([2, \infty)\). This implies:

• The set includes every number from 2 and upwards.
• Infinity is never reached or included, hence the parenthesis.

Interval notation simplifies complex solutions into quick and easy symbols, making them useful for conveying mathematical ideas efficiently.

Solving Compound Inequalities

Solving compound inequalities involves breaking the task into manageable steps. Understanding compound inequalities starts with figuring out the individual inequalities' solutions.
Let's walk through the process:

• Start with each inequality separately. Here, the first was \(5(x-2) \geq 0\), solved to \(x \geq 2\).
• The next, \(-3x < 9\), when solved gives \(x > -3\). Remember, dividing by a negative flips the inequality symbol.
• Combine these into a single solution using "and," which means both conditions must be true simultaneously.
• Thus, \(x \geq 2\) becomes the gating condition—the stricter of the two—since any \(x\) that satisfies this also satisfies \(x > -3\).

Grasping compound inequalities involves solving each part, then analyzing the overlap to present a coherent solution. The discipline in each step helps create a strong foundation for more complex algebraic work.