Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 21-70: Solve the inequality, and express the solutions in terms of intervals whenever possible. $$ 2 x+5<3 x-7 $$

Step-by-Step Solution

Answer

The solution in interval notation is \((12, \infty)\).

Step 1: Isolate Terms with x

To solve the inequality, begin by moving all terms involving \(x\) to one side of the inequality. We start by subtracting \(2x\) from both sides: \[2x + 5 - 2x < 3x - 7 - 2x\] which simplifies to \[5 < x - 7\], since \(2x - 2x = 0\) and \(3x - 2x = x\).

Step 2: Isolate x

Next, we will isolate \(x\) by adding 7 to both sides of the inequality to cancel out the \(-7\): \[5 + 7 < x - 7 + 7\] which simplifies to \[12 < x\].

Step 3: Express in Interval Notation

Now that we know \(x\) must be greater than 12, we can express the solution in interval notation as \((12, \infty)\). This means that \(x\) can be any real number greater than 12.

Key Concepts

Interval notation made simple

Interval notation is a compact way to describe a set of numbers, commonly used when working with inequalities. The solution to an inequality can often be a range, not just a single number. Interval notation helps express this range clearly and efficiently.
For example, if we find that a solution involves all numbers greater than 12, we write this as \((12, \infty)\). The parentheses, \(()\), indicate that the endpoints are not included in the solution. This means 12 is not a part of the solution set, only numbers strictly greater than 12.

• \((a, b)\) is used for numbers between but not including \(a\) and \(b\).
• \([a, b]\) includes the numbers \(a\) and \(b\).
• \((a, b]\) includes all numbers from \(a\) to \(b\), but not \(a\).
• \([a, \infty)\) includes \(a\) and all numbers greater than \(a\).

This set of rules provides a consistent method to express many types of solutions for inequalities.

Understanding isolation of variables

Isolation of variables is a fundamental step in solving inequalities, as it involves rearranging terms in an equation to get the variable by itself on one side. This process helps us understand what range of values the variable can take to satisfy the inequality.
Consider the inequality: \(2x + 5 < 3x - 7\). Our goal is to isolate \(x\). We do this by:

• First, moving all terms with \(x\) to one side. This is done by subtracting \(2x\) from both sides, resulting in \(5 < x - 7\).


• Next, adjust the equation further by adding 7 to both sides to eliminate constants from the \(x\) side, giving \(12 < x\).

Now, \(x\) is isolated. This step-by-step process unveils the relation between the values and allows us to see the full solution set when expressed in interval notation.
Proper isolation is key to simplifying expressions and can make more complex problems manageable.

Grasping linear inequalities

Linear inequalities are like linear equations, but instead of an equality, they have an inequality sign such as <, \(>\), \(\leq\), or \(\geq\). Solving them involves finding which values of the variable make the inequality true. This can often be done using similar strategies as those used in solving equations.
For instance, solving the inequality \(2x + 5 < 3x - 7\) requires the same principles as solving the linear equation \(2x + 5 = 3x - 7\), except the focus is on preserving the inequality sign. We carefully perform operations on both sides:

• Subtract \(2x\) from both sides to simplify to \(5 < x - 7\).
• Add or subtract constants as needed to isolate \(x\), leading to \(12 < x\).

Linear inequalities often result in a range of solutions, illustrating why interval notation is useful. Unlike linear equations, solutions are not a single point but rather a continuous set of values representing all possible solutions.
This nature makes understanding and solving linear inequalities an essential skill in algebra.