Problem 13 - Calculus: One and Several Variables

Question

Solve the inequality. Express the solution as an interval or as the union of intervals. Mark the solution on a number line. $$5 x-2<0$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x < 0.4\), which can be expressed as the interval \(-\infty, 0.4\), excluding 0.4. On the number line, this is represented by shading to the left of 0.4.

Step 1: Isolate x

Start by adding 2 to both sides to isolate the term that contains x: \(5x - 2 + 2 < 0 + 2 \rightarrow 5x < 2.\) Now, to fully isolate x, divide both sides by 5: \(\frac{5x}{5} < \frac{2}{5} \rightarrow x < 0.4.\)

Step 2: Express The Solution As An Interval

The solution of the inequality \(x < 0.4\) can be expressed as the interval \(-\infty, 0.4\), excluding 0.4 because the inequality does not have 'or equal to'.

Step 3: Mark the Solution on a Number Line

Draw a number line, mark 0.4 on it and draw an open circle over 0.4. The solutions are all numbers less than 0.4, so shade the number line to the left of 0.4.

Key Concepts

Isolation of Variables

Solving inequalities often begins with isolating the variable on one side of the inequality. This means getting the variable alone on one side to simplify the expression. Consider the inequality:\[5x - 2 < 0\]To isolate \(x\), start by eliminating the constant on the same side. Add 2 to both sides:\[5x - 2 + 2 < 0 + 2 \rightarrow 5x < 2\]Next, divide both sides by 5 to solve for \(x\):\[\frac{5x}{5} < \frac{2}{5} \rightarrow x < 0.4\]By expressing the inequality this way, we make it simpler to understand which values \(x\) can take. Always remember that reversing the inequality sign is necessary when multiplying or dividing by a negative number, though this step didn't apply here.

Interval Notation

Once the variable is isolated, the next step is to express the solution in interval notation. This notation is an efficient way to describe a range of values. For the inequality \(x < 0.4\), the solution set is all numbers less than 0.4.Use parentheses to denote that an endpoint is not included. So, the solution in interval notation is:\((-\infty, 0.4)\)This tells us that the solution includes all real numbers from negative infinity to 0.4, but not including 0.4 itself.

Graphical Representation

Graphically representing solutions helps visualize what solutions look like. Using a number line is a common method.Place a circle at the point 0.4 on the number line to represent that 0.4 is not included. This is called an open circle, indicating that the value itself is not part of the solution.Shade or draw a line extending to the left of 0.4, covering all numbers less than 0.4. This visually displays all possible solutions of the inequality \(x < 0.4\). The combination of markers and shading offers a clear image of where solutions lie.

Number Line Concepts

Understanding how to use a number line is crucial in algebra and inequalities. On a number line, numbers increase to the right and decrease to the left.Each point on a number line corresponds to a real number. When marking solutions, it's important to use open circles for inequalities like \(<\) or \(>\) and closed circles for \(\leq\) or \(\geq\).For inequality solutions, shading indicates where the solution set lies—on one side of a point or a segment of the number line. This creates an intuitive way to see the possible values \(x\) may take based on the inequality's constraints.