Problem 10 - College Algebra

Question

In Exercises 1-12, graph the solutions of each inequality on a number line. $$-3 \leq x<7$$

Step-by-Step Solution

Answer

The solution for the inequality \(-3 \leq x<7\) includes values of \(x\) that are equal to or greater than -3, but less than 7. This is represented on the number line with a filled circle at -3 and an open circle at 7, with a line drawn between them.

Step 1: Interpret the Inequality

Understand what the inequality \(-3 \leq x<7\) is stating. It includes all values of \(x\) that are equal to or greater than -3 and less than 7.

Step 2: Draw a number line

Sketch a simple number line, include numbers from -4 to 8 to provide enough context for the solution.

Step 3: Mark the interval on the number line

The interval from -3 to 7 needs to be marked on the number line. Start by marking a filled circle (which indicates that -3 is included) at -3 and an empty circle (which indicates 7 is excluded) at 7.

Step 4: Highlight the solution on the number line

Draw a line (or highlight using color) between -3 and 7 on the number line to show the range of values \(x\) that satisfy the inequality.

Key Concepts

Number Line Solutions

When you're faced with an inequality like \( -3 \leq x < 7 \) and asked to graph the solutions on a number line, you are essentially mapping out all the possible values of \(x\) that make the inequality true.

A number line is a visual tool that helps you understand the range of numbers included in the solution. For our given inequality, the number line will show every point between -3 and 7, including -3 but not including 7. The filled circle on -3 signifies that -3 is a solution (since the inequality symbol \(\leq\) means 'less than or equal to'), whereas the empty circle on 7 indicates that 7 is not a solution (the \(<\) symbol stands for 'less than').

Graphing on a number line makes it easier to picture which values are part of the solution set and assists in eliminating any confusion about boundary values, especially when using symbols like \(\leq\) and \(<\).

Solving Inequalities

Solving inequalities is a fundamental skill in algebra that allows you to find the set of all possible values that satisfy a given condition. Unlike an equation which typically has one or two solutions, an inequality usually has a range of solutions.

To solve the inequality \( -3 \leq x < 7 \), you interpret each part of the inequality. The part \( -3 \leq x \) tells us that x is greater than or equal to -3 and the part \( x < 7 \) tells us x is less than 7. The solution is the overlap of both conditions—every number that is both greater than or equal to -3 and less than 7.

Check Your Solution

Always verify your answers. For instance, if you plug in a value like 0, which lies between -3 and 7, it should satisfy the inequality. If you test a number outside the range, like -4 or 8, those should not make the inequality true. This checking process is an excellent way to ensure confidence in your solutions.

Interval Notation

Interval notation is a concise way of writing the set of solutions to an inequality. It's quite useful in clearly defining the exact numbers that are included in the solution and is commonly preferred in higher mathematics due to its brevity and precision.

The inequality \( -3 \leq x < 7 \) would be written in interval notation as \( [-3, 7) \). The square bracket \([\) means that -3 is included in the set—it's 'closed' to -3—while the parenthesis \()\) indicates that 7 is not included—'open' at 7. This notation allows anyone to quickly understand the range of acceptable values without the need for drawing a number line.

Advantages of Interval Notation

Interval notation is particularly advantageous when working with unions or intersections of sets or when dealing with complex, compounded inequalities. It also helps to avoid ambiguity in conveying which endpoints are part of the solution set.