Problem 29 - Algebra for College Students

Question

Solve each compound inequality. Graph the solution set, and write it using interval notation. $$ 3 x-4 \leq 8 \text { and }-4 x+1 \geq-15 $$

Step-by-Step Solution

Answer

The solution set is \((-\infty, 4]\).

Step 1: Solve the first inequality

We start with the inequality \(3x - 4 \leq 8\). First, add 4 to both sides to get \(3x \leq 12\). Then, divide both sides by 3 to find \(x \leq 4\).

Step 2: Solve the second inequality

Consider the second inequality \(-4x + 1 \geq -15\). First, subtract 1 from both sides to get \(-4x \geq -16\). Then, divide both sides by -4, remembering to reverse the inequality sign, resulting in \(x \leq 4\).

Step 3: Find the overlapping solution set

The solution to the compound inequality consists of values of x that satisfy both inequalities. Since both inequalities result in \(x \leq 4\), the combined solution is \(x \leq 4\).

Step 4: Write the solution in interval notation

Since \(x \leq 4\), the solution in interval notation is \((-\infty, 4]\).

Step 5: Graph the solution set

On a number line, shade the region to the left of 4 and include the point at 4 with a solid dot, indicating that 4 is part of the solution set.

Key Concepts

Interval Notation

Interval notation is a concise way to describe a set of numbers along a number line. It uses intervals to show the starting and ending points.
For example, if the solution set to an inequality is all numbers less than or equal to 4, this is written as \((-\infty, 4]\).
The round bracket \( ( \) means that ∞ is not included.
The square bracket \( [ \) means that 4 is included.
It's a useful shorthand that helps us quickly communicate which numbers are in the solution set.

Inequality Solution

Solving an inequality means finding all the values of the variable that make the inequality true.
Consider solving the inequality \( -4x + 1 \geq -15 \):

• First, subtract 1 from both sides: \( -4x \geq -16\)
• Then, divide both sides by -4, remembering to reverse the inequality: \( x \leq 4 \)

We need to be careful about reversing the inequality sign when we divide by a negative number.
The solution tells us that any value of x less than or equal to 4 satisfies the inequality.

Graphing Inequalities

Graphing inequalities involves showing the solution set on a number line.
To graph \( x \leq 4 \) on a number line:

• Draw a darkened heavy dot at 4 to indicate that 4 is included (since it's \( \leq \)
• Shade the number line to the left of 4 to represent all values less than 4

This visual representation makes it easy to understand at a glance which numbers satisfy the inequality.
Graphing can be helpful in verifying solutions or just conveying the answer in a clear and simple form.