Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ -3<1-6 x \leq 4 $$

Short answer

The solution in interval notation is \([-\frac{1}{2}, \frac{2}{3})\).

Step-by-step solution

Move Constant Term

First, we need to isolate the term with the variable. Start by subtracting 1 from all parts of the inequality:\[-3 - 1 < 1 - 1 - 6x \leq 4 - 1\]This simplifies to:\[-4 < -6x \leq 3\]

Divide by Coefficient

Next, we need to solve for \(x\). Since \(-6\) is the coefficient of \(x\), divide the entire inequality by \(-6\). Remember, dividing by a negative number reverses the inequality signs:\[\frac{-4}{-6} > x \geq \frac{3}{-6}\]This simplifies to:\[\frac{2}{3} > x \geq -\frac{1}{2}\]

Write in Interval Notation

Now, express the solution set in interval notation. The inequality states \(-\frac{1}{2} \leq x < \frac{2}{3}\). In interval notation, this is written as:\[[-\frac{1}{2}, \frac{2}{3})\]

Sketch the Graph

On a number line, mark the points \(-\frac{1}{2}\) and \(\frac{2}{3}\). Draw a closed circle at \(-\frac{1}{2}\) to indicate that \(x\) can equal \(-\frac{1}{2}\) and an open circle at \(\frac{2}{3}\) to show that \(x\) cannot equal \(\frac{2}{3}\). Shade the region between these two points to represent the solution set.

Key concepts

Interval Notation

Interval notation offers a concise way to express ranges of values that satisfy an inequality. When dealing with inequalities, we often arrive at a solution set that includes a range of numbers. In the exercise, we derived that the inequality \(-3 < 1 - 6x \leq 4\) simplifies down to \(-\frac{1}{2} \leq x < \frac{2}{3}\).

This means \(x\) can take any value from \(-\frac{1}{2}\) up to, but not including, \(\frac{2}{3}\). In interval notation, this is succinctly written as \([-\frac{1}{2}, \frac{2}{3})\).

• A square bracket, \([\), means the endpoint is included, i.e., \(x\) can equal \(-\frac{1}{2}\).
• A round parenthesis, \(()\), means the endpoint is not included, so \(x\) cannot equal \(\frac{2}{3}\).

Interval notation is a powerful tool because it clearly states the starting and stopping points of solutions and whether those points are included or not, all in a compact format.

Graphical Representation

Graphically representing the solution of an inequality helps us visualize which values of \(x\) satisfy the inequality. For the inequality \(-\frac{1}{2} \leq x < \frac{2}{3}\), we use a number line to give a clear representation.

When sketching this on a number line:

• We place a closed circle at \(-\frac{1}{2}\) because this value is included in the solution (\(x\) can be equal to \(-\frac{1}{2}\)).
• We place an open circle at \(\frac{2}{3}\) to indicate that this endpoint is not included (\(x\) cannot be \(\frac{2}{3}\)).
• The number line between \(-\frac{1}{2}\) and \(\frac{2}{3}\) is shaded to show all values in this range are solutions.

Graphical representations are essential as they offer a visual confirmation of the solution set, making it easier to understand for those who are more visually inclined. They also help in showing the extent and nature of the range clearly.

Solving Linear Inequalities

Solving linear inequalities is a fundamental skill that involves finding all possible solutions for a variable that makes an inequality true. The process starts with isolating the variable just as in algebraic equations, but it includes an important consideration: if you multiply or divide by a negative number, you must reverse the inequality symbols.

In our example, we began by simplifying \(-3 < 1 - 6x \leq 4\) to isolate the \(x\) term:

• First, we subtracted 1 from all parts of the inequality to maintain balance, resulting in \(-4 < -6x \leq 3\).
• Second, we divided through by \(-6\), reversing the inequality signs to get \(\frac{2}{3} > x \geq -\frac{1}{2}\).

The rules of inequalities require careful handling to ensure accuracy in results. Always remember to flip the inequality when multiplying or dividing by a negative, as failing to do so can lead to incorrect solutions. Practice in these methods solidifies understanding and builds confidence in solving a wide range of linear inequalities effectively.