Problem 33 - Algebra for College Students

Question

Express each interval as an inequalit using the variable \(x\). For example, we can express the inter val \([5, \infty)\) as \(x \geq 5\). \(15<1-7 x\)

Step-by-Step Solution

Answer

The inequality is \(x < -2\).

Step 1: Subtract 1 from Both Sides

To isolate the variable term, subtract 1 from both sides of the inequality. This gives you the equation:\[15 - 1 < -7x\]Simplifying the left side gives us:\[14 < -7x\]

Step 2: Divide Both Sides by -7

Since we want to solve for \(x\), divide both sides of the inequality by \(-7\) to isolate \(x\). Remember, when you divide or multiply an inequality by a negative number, you must reverse the inequality sign.\[\frac{14}{-7} > x\]Simplifying the left side gives us:\[-2 > x\]

Step 3: Rewrite the Inequality

We usually write inequalities with the variable on the left side. Rewrite the inequality with \(x\) on the left side. This gives us:\[x < -2\]

Key Concepts

Interval Notation

Interval notation is a concise way to express a range of values on a number line. It shows the set of numbers that satisfy an inequality.
For example, the interval \([-2, 5]\) includes all numbers between -2 and 5, including -2 and 5 themselves.
\((-2, 5)\) would mean all numbers between -2 and 5, but not the endpoints.

• The square bracket [ or ] means the number is included (closed interval).
• The round parenthesis ( or ) means the number is not included (open interval).

In our exercise, the solution \(x < -2\) can be expressed in interval notation as \((-\infty, -2)\). This interval means any number smaller than -2, but not including -2 itself. The infinity symbol is always paired with a parenthesis since it is conceptually unreachable.

Solving Inequalities

Solving inequalities involves similar steps to solving equations, with a key difference. For inequalities, the relationship between the quantities is not always '='. It could be '<', '>', '\(\leq\)', or '\(\geq\)'.
To solve inequalities correctly, follow these steps:
- **Identify the inequality**: Determine which type of inequality you're dealing with.
- **Isolate the variable**: Use operations such as addition, subtraction, multiplication, or division to get the variable of interest by itself on one side of the inequality.
- **Reverse the inequality sign if necessary**: Crucially, when multiplying or dividing both sides of the inequality by a negative number, you must reverse the inequality sign to maintain the truth of the statement.
In the exercise provided, this very rule is applied in Step 2, transforming \(-7x < 14\) into \(x > -2\) upon dividing by -7.

Inequality Properties

Inequality properties help us understand the behavior and relationship of numbers when they are not equal.
These properties are:

• **Addition and Subtraction**: You can add or subtract the same quantity from both sides without changing the sign of the inequality. For example, \(a < b\) gives us \(a + c < b + c\).
• **Multiplication and Division**: Multiplying or dividing both sides of an inequality by a positive number keeps the inequality sign the same. However, if you multiply or divide by a negative number, the inequality sign must be reversed to maintain the correct relationship.
• **Transitive Property**: If \(a < b\) and \(b < c\), then \(a < c\). This property helps chain together multiple inequalities.

Understanding these properties is crucial, especially in our example where dividing by a negative number was key to finding the correct solution.