Problem 54 - Elementary and Intermediate Algebra

Question

For each of the following, write a second inequality with the same meaning. $$ a>9 $$

Step-by-Step Solution

Answer

a - 9 > 0

Step 1: Understand the Original Inequality

Start by analyzing the given inequality which is stated as:a > 9This inequality says that 'a' is greater than 9.

Step 2: Identify an Equivalent Inequality

To write a second inequality with the same meaning, think of other ways to express that 'a' is greater than 9. One common approach is to subtract 9 from both sides of the inequality.

Step 3: Rewrite the Inequality

Subtract 9 from both sides of the inequality:a - 9 > 0This is the same as the original inequality because if 'a' is greater than 9, then subtracting 9 from 'a' will result in a number greater than 0.

Step 4: Verify the Equivalent Inequality

Check if the new inequality reflects the same relationship:If 'a' is greater than 9, then 'a-9' must be greater than 0.Thus, the new inequality a - 9 > 0 has the same meaning as the original inequality.

Key Concepts

equivalent inequalities

To understand equivalent inequalities, let's first break down the term 'equivalent'. It means 'equal in value, amount, function, meaning, etc.'.
When we talk about equivalent inequalities in algebra, we refer to different expressions that describe the same relationship between quantities. For example, the inequality given in the problem, \(a > 9\), can be rewritten in another form without changing its meaning. By performing the same mathematical operation on both sides of the inequality, we maintain its truth.
After subtracting 9 from both sides, the inequality becomes \(a - 9 > 0\). This is an equivalent inequality because it conveys the same idea: 'a' is greater than 9.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations. In this exercise, the given inequality \(a > 9\) and its equivalent form \(a - 9 > 0\) are examples of algebraic expressions.
To form algebraic expressions, remember that:

• Variables represent unknown quantities and are usually denoted by letters (e.g., \(a\)).
• Constants are fixed values (e.g., 9).
• Mathematical operations include addition, subtraction, multiplication, and division.

Understanding how to manipulate these elements using algebraic rules is essential for solving inequalities and other algebraic problems.

solving inequalities

Solving inequalities involves finding the values of the variable that make the inequality true. The process is very similar to solving equations, with some important differences.
Here are key steps to solve inequalities:

• Perform the same mathematical operation on both sides to keep the inequality balanced. For instance, subtracting 9 from \(a>9\) gave us \(a-9>0\).
• Keep the inequality sign pointing in the correct direction when multiplying or dividing by a positive number.
• Reverse the inequality sign if you multiply or divide by a negative number.

Once solved, you can represent the solutions on a number line or with interval notation to visualize the range of possible values.

greater than inequalities

Greater than inequalities, such as \(a > 9\), indicate that the variable must be larger than a certain value. These inequalities open to the right, showing that all values larger than the given number make the statement true.
Some properties of greater than inequalities include:

• If an inequality states \(a > 9\), any value of 'a' from 9.1 to infinity satisfies it.
• Operations modifying the inequality (like subtracting 9 in \(a > 9\)) will transform it but keep the 'greater than' relationship (resulting in \(a - 9 > 0\)).
• When solving, ensure that the variable remains on one side, clearly indicating the range of possible values.

Understanding how to work with greater than inequalities helps in solving complex problems and finding valid solutions.