Problem 10 - Elementary and Intermediate Algebra

Question

Fill in the blanks. If we multiply both sides of an inequality by a negative number, the direction of the inequality must be _____ for the inequalities to have the same solutions.

Step-by-Step Solution

Answer

reversed

Step 1: Understanding Inequalities

When you have an inequality, it consists of two sides separated by an inequality symbol such as <, >, ≤, or ≥. For example, in the inequality \( x < 5 \), \( x \) is less than 5.

Step 2: Effect of Multiplying by a Positive Number

If you multiply or divide both sides of an inequality by a positive number, the direction of the inequality remains the same. For example, multiplying both sides of \( x < 5 \) by 2 results in \( 2x < 10 \).

Step 3: Effect of Multiplying by a Negative Number

If you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality to maintain its truth. For example, multiplying both sides of \( x < 5 \) by -1 results in \( -x > -5 \), reversing the inequality symbol to maintain the correct solution set.

Step 4: Identifying the Change Needed

Given the concept learned from multiplying by a negative number, the blank in the exercise needs something that indicates a reverse in direction.

Key Concepts

Multiplying Inequalities

When working with inequalities, one critical rule to remember is how different operations affect them. Multiplying both sides of an inequality is a common operation you might encounter. Here are some important points to keep in mind:

• If you multiply both sides by a positive number, the inequality remains unchanged. For instance, multiplying both sides of the inequality \( x < 3 \) by 2 gives \( 2x < 6 \).
• If you multiply both sides by zero, it can make the inequality meaningless since multiplying by zero gives a result of zero on both sides.
• If you multiply by a negative number, you must flip the inequality symbol. This flip ensures the inequality still holds true. For example, if you multiply \( x < 3 \) by \(-1\), you need to flip the inequality to \(-x > -3\).

Understanding these rules helps you solve inequalities correctly.

Negative Numbers

Negative numbers can often change the behavior of mathematical statements, especially inequalities. They create an interesting twist by altering how inequalities must be handled. Let's take a closer look:

• When you multiply or divide an inequality by a negative number, flipping the inequality symbol is necessary. This is because negative multiplication reverses the order between the numbers. For example, while 3 is less than 5, \(-3\) becomes greater than \(-5\).
• This reversal might feel counterintuitive, but it helps preserve the true relationship between the expressions on each side of the inequality.
• Never forget this step, as leaving the inequality direction incorrect can lead to wrong answers.

Keeping the rules for negative numbers in mind can prevent common mistakes when solving inequalities.

Inequality Symbols

Inequality symbols are at the heart of understanding inequalities. Knowing what these symbols mean and how they behave is key.

• The primary symbols you will encounter are \(<\), \(>\), \(\leq\) (less than or equal to), and \(\geq\) (greater than or equal to).
• These symbols define the relationship between values or expressions in an inequality. For instance, \(x < y\) indicates that \(x\) is less than \(y\).
• Understanding when and why these symbols reverse when multiplying or dividing by negative numbers is crucial. This reversal maintains the inequality's truth.

Gaining a solid grasp of inequality symbols and their proper use allows you to accurately and confidently solve inequality problems.