Question

Sketch the graph of the inequality. $$-5 x>-10$$

Short answer

The solution of the inequality is \(x < 2\). On a number line, represent this by drawing an open circle at \(x = 2\) and shading all points to the left.

Step-by-step solution

- Solve the inequality

This is a simple inequality that can be solved by dividing both sides with -5. Remember that when we divide by a negative number, the direction of the inequality changes. So upon dividing by -5, the inequality becomes \(x < 2\)

- Graph the solution

Draw a number line and mark the point \(x = 2\) on that line. Since the solution of the inequality is \(x < 2\), we are looking for the set of points that are to the left of 2. It contains every number less than 2, so it should be emphasized that the point at \(x = 2\) does not belong to the solution set. So draw an open circle at \(x = 2\) and shade to the left to represent the solution set.

Key concepts

Solving Inequalities

When it comes to solving inequalities, the process often resembles solving a regular equation. However, a key difference occurs when multiplying or dividing both sides by a negative number – the inequality symbol must be flipped. This is a crucial step that distinguishes inequalities from equations. For instance, solving the inequality \( -5x > -10 \) involves dividing both sides by -5, which flips the inequality to \( x < 2 \). Understanding this rule ensures accurate solutions to inequalities and sets the foundation for correctly graphing the solution on a number line.

To avoid losing track of this important step, you might find it helpful to have a personal checklist when dealing with inequalities. Make sure that one item on that checklist prompts you to reverse the inequality if you are dividing or multiplying by a negative number. This practice will help improve the accuracy of solving these mathematical expressions.

Number Line Representation

Visualizing inequalities on a number line brings clarity to solutions that are sometimes not immediately intuitive. A number line is a straight line with numbers placed at equal intervals along its length. In representing an inequality like \( x < 2 \), a number line helps to show all the possible values that \( x \) can take. For this particular inequality, you would:

• Draw a horizontal line and mark increments to represent numbers.
• Locate and label the point that corresponds to \( x = 2 \).
• Since the inequality indicates \( x \) is less than 2, draw an open circle at 2 to show that it is not included in the solution set.
• Shade the line to the left of the open circle to represent all the numbers less than 2 that satisfy the inequality.

Using these steps, the concept of a solution set becomes more apparent as the shaded region on the number line.

It's beneficial to think of the number line as a visual story of the inequality. The open or closed circle tells us if a point is part of the solution or not, while the shaded part narrates the range of viable solutions. By improving the way we represent inequalities on number lines, we enhance our understanding and communication of possible solutions.

Inequality Symbols

The symbols used in inequalities convey the limits and relationships between numbers. In basic terms, these symbols include:

• \(<\) which means 'less than',
• \(>\) which means 'greater than',
• \(\geq\) which means 'greater than or equal to',
• \(\leq\) which means 'less than or equal to'.

An inequality symbol points in the direction of the smaller value and opens towards the larger value. It's critical to recognize what each inequality symbol indicates to understand the range of the solution clearly.

For example, in the inequality \( x < 2 \) solved earlier, the '<' symbol indicates that \( x \) is less than 2, excluding the number 2 itself. When graphing, this is why an open circle is used. If the inequality were \( x \leq 2 \) instead, a closed circle would be used on the number line, because 2 would be included as part of the solutions. Mastering these symbols is like learning a new language that is fundamental to advancing in mathematics, especially when expressing and interpreting solutions.