Problem 59 - Algebra A Combined Function

Question

Six more than twice a number is greater than negative fourteen. Find all numbers that make this statement true.

Step-by-Step Solution

Answer

The numbers greater than -10 make the statement true.

Step 1: Translate the problem into an inequality

The statement "six more than twice a number is greater than negative fourteen" can be translated into a mathematical inequality. Let the number be represented by \(x\). Then, twice the number is \(2x\). Adding six to this, we get the inequality: \(2x + 6 > -14\).

Step 2: Isolate the variable term

To solve the inequality \(2x + 6 > -14\), you first need to isolate the term with \(x\) on one side. Start by subtracting 6 from both sides:\[2x + 6 - 6 > -14 - 6\]which simplifies to:\[2x > -20\].

Step 3: Solve for the variable

To isolate \(x\), divide both sides of the inequality by 2:\[\frac{2x}{2} > \frac{-20}{2}\]This simplifies to:\[x > -10\].

Step 4: Interpret the solution

The solution \(x > -10\) means that all numbers greater than -10 will satisfy the original inequality. Therefore, any number larger than -10 makes the inequality statement true.

Key Concepts

Mathematical Translation

When solving problems involving inequalities, the first crucial step is transforming the verbal statement into a mathematical expression. This process is called mathematical translation.
In our problem, the phrase "six more than twice a number is greater than negative fourteen" needs to be interpreted mathematically.

• "A number" is typically represented by a variable, often denoted as \(x\).
• "Twice a number" means multiplying the variable by 2, expressed as \(2x\).
• Adding "six more" translates to \(2x + 6\).
• "Is greater than negative fourteen" becomes the inequality \(> -14\).

These steps let us write the inequality as \(2x + 6 > -14\). Translating word problems using precise mathematical language is the foundation for effective problem-solving.

Solving Inequalities

Once an inequality is established, the next step is solving it to find which values satisfy it. In our example, we have the inequality \(2x + 6 > -14\).
Here’s how you can solve it step-by-step:

• Isolate the variable term: Start by eliminating constants added to the variable term. For \(2x + 6 > -14\), subtract 6 from both sides to get \(2x > -20\).
• Divide to solve for the variable: To make \(x\) the subject, divide every term by 2, resulting in \(x > -10\).

When solving inequalities, the operations of adding, subtracting, multiplying, and dividing are used similarly to solving equations, with one exception: if you multiply or divide by a negative number, the inequality sign should be reversed. This step is not needed here, as we dealt with a positive divisor.

Variables in Algebra

Variables are symbols, usually letters, that represent numbers or values in algebraic expressions and equations.
In our problem, the variable \(x\) stands for an unknown number that we are trying to determine. Here's why they are important:

• Variables allow us to manipulate and solve equations or inequalities to find unknown values. They provide flexibility in representing unknown quantities.
• By using variables, we can create general solutions applicable to numerous problems, not just a single instance.

In our example, we note that \(x > -10\) means any number greater than \(-10\) will satisfy the inequality. Thus, variables serve as a powerful tool in algebra, enabling us to express complex ideas and solve diverse problems efficiently.