Problem 15 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(4 x<-16\)

Step-by-Step Solution

Answer

The solution is \( x < -4 \), graphed with an open circle at -4 and shaded to the left, written as \( (-\infty, -4) \).

Step 1 - Isolate the variable

To solve the inequality, start by isolating the variable on one side of the inequality. Divide both sides by 4: \[ \frac{4x}{4} < \frac{-16}{4} \] Simplifying this, we get: \[ x < -4 \]

Step 2 - Graph the solution

To graph the solution, find -4 on the number line. Because the inequality is 'less than', draw an open circle at -4 and shade the area to the left to show all numbers less than -4.

Step 3 - Write the solution in interval notation

The interval notation for all numbers less than -4 is written as: \( (-\infty, -4) \)

Key Concepts

Isolate the Variable

When solving inequalities, a key step is to isolate the variable. This means we want to get the variable by itself on one side of the inequality symbol. In our example, the inequality is \(4x < -16\). Here’s how you do it:

• Divide both sides of the inequality by 4.
• Remember, when you divide or multiply both sides of an inequality by a positive number, the direction of the inequality sign stays the same.

So, \( \frac{4x}{4} < \frac{-16}{4} \) simplifies to \( x < -4 \). You've successfully isolated the variable!

Graphing Solutions

Graphing solutions for inequalities helps you visualize the range of possible values for the variable. Here’s how to graph \( x < -4 \):

• Draw a number line.
• Locate -4 on the number line. Because the inequality is 'less than' and not 'less than or equal to', we use an open circle at -4 to show that -4 is not included in the solution.
• Shade the area to the left of -4. This shading means all numbers less than -4 are solutions to the inequality.

Open circles are used when the variable is strictly less than (\( < \)) or greater than (\( > \)), while closed circles are used for 'less than or equal to' (\( \leq \)) or 'greater than or equal to' (\( \geq \)).

Interval Notation

Interval notation is a concise way to write the set of solutions for an inequality. For our example \( x < -4 \), the interval includes all numbers less than -4.

• The smallest part of the interval is negative infinity (\( -\infty \)). Since there is no number smaller than negative infinity, it’s always used to represent unbounded lower limits.
• The largest part of our interval is -4. Since -4 is not included in the solution (as indicated by the open circle in the graph), we use a parenthesis (\( ( \)). If it were included, we would use a bracket (\( [ \)).

So, our interval notation for \( x < -4 \) is \( (-\infty, -4) \). This notation indicates that the solution set includes all numbers less than -4, but not -4 itself.