Question

Express the solution set of the given inequality in interval notation and sketch its graph. $$ 4<5-3 x<7 $$

Short answer

The solution set is \((-\frac{2}{3}, \frac{1}{3})\).

Step-by-step solution

Solve the First Part of Inequality

First, we solve the left part of the compound inequality. We have: \[ 4 < 5 - 3x \] Subtract 5 from both sides to isolate the term with \( x \): \[ -1 < -3x \] Now, divide through by \(-3\). Remember to flip the inequality sign when dividing by a negative:\[ \frac{-1}{-3} > x \] Simplifying gives:\[ \frac{1}{3} > x \] or equivalently:\[ x < \frac{1}{3} \]

Solve the Second Part of Inequality

Next, we solve the right part of the compound inequality. We have:\[ 5 - 3x < 7 \]Subtract 5 from both sides:\[ -3x < 2 \] Divide through by \(-3\). Remember to flip the inequality sign:\[ x > \frac{-2}{3} \]

Combine the Solutions

Now, we combine the solutions from Step 1 and Step 2. The solution set is:\[ \frac{-2}{3} < x < \frac{1}{3} \]This means \( x \) is greater than \(-\frac{2}{3}\) and less than \(\frac{1}{3}\).

Express in Interval Notation

To express the solution set in interval notation:\[ (-\frac{2}{3}, \frac{1}{3}) \]This interval represents all \( x \) values between \(-\frac{2}{3}\) and \(\frac{1}{3}\), excluding the endpoints because the inequality is strict.

Sketch the Graph

On a number line, plot points at \(-\frac{2}{3}\) and \(\frac{1}{3}\). Use open circles at these points to indicate that these are not included in the interval. Shade the region between these open circles to represent all values of \( x \) that satisfy the inequality.

Key concepts

Interval Notation

Interval notation is a convenient way to represent the set of all numbers between two endpoints. In this context, it's used to express the solution of our compound inequality. Interval notation uses brackets to denote whether endpoints are included or not.

For example, the interval \((-\frac{2}{3}, \frac{1}{3})\) indicates all numbers between \(-\frac{2}{3}\) and \(\frac{1}{3}\), excluding the endpoints themselves. Parentheses are used because the inequality does not include the values at \(-\frac{2}{3}\) and \(\frac{1}{3}\).

This contrasts with the use of square brackets, \[a, b\], which indicate that the interval includes the endpoints \(a\) and \(b\). Remembering the difference between open and closed intervals is essential when writing solutions in interval notation.

Number Line Graph

A number line graph visually represents solutions to inequalities. It showcases which values make the inequality true. Imagine a straight line with evenly spaced numbers. Now, plot points at key locations to show the interval or range of numbers that are part of your solution.

In our solution, the graph includes points at \(-\frac{2}{3}\) and \(\frac{1}{3}\). However, we use open circles at these points because they're not part of the solution set due to the inequality being strict. \((-\frac{2}{3}, \frac{1}{3})\) means we don't include these endpoints.

Shade the number line between \(-\frac{2}{3}\) and \(\frac{1}{3}\), representing that all numbers in this range satisfy the inequality. A number line graph is helpful because it offers a clear visual sense of which values work, enhancing your understanding of the inequality solution.

Inequality Solution

Solving an inequality means finding all possible values of a variable that make the inequality true. In our exercise, we are dealing with a compound inequality, which involves two separate inequalities joined by 'and'.

First, let's solve the part \(4 < 5 - 3x\). After isolating \(x\), we find \(x < \frac{1}{3}\). Next, solve \(5 - 3x < 7\) to find \(x > -\frac{2}{3}\).

Combining these results forms the overall inequality solution: \(-\frac{2}{3} < x < \frac{1}{3}\). This indicates that \(x\) must be both greater than \(-\frac{2}{3}\) and less than \(\frac{1}{3}\) to satisfy both inequality parts, highlighting compound inequality's logic.

Step-by-Step Solution

Approaching a problem step-by-step ensures you cover each requirement completely without errors. Here, breaking down the compound inequality into simpler parts is key.

1. **Solve separately**: Tackle each side of the compound inequality. Isolate \(x\) in each inequality. - For \(4 < 5 - 3x\), simplify to \(x < \frac{1}{3}\). - For \(5 - 3x < 7\), simplify to \(x > -\frac{2}{3}\). 2. **Combine results**: This gives you a clearer picture of \(x\)'s valid range, \(-\frac{2}{3} < x < \frac{1}{3}\).3. **Translate to interval notation**: Since both solutions must occur simultaneously, convert to \((-\frac{2}{3}, \frac{1}{3})\).4. **Graph on number line**: Plot with open circles and shade to visually confirm the solution.By following a detailed step-by-step method, you ensure accurate and understandable solutions, which is crucial in dealing with inequalities.