Problem 51 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ 2 x-1

Step-by-Step Solution

Answer

Answer: The solution to the inequality is \(x < 6\). This means any value of x less than 6 will satisfy the given inequality.

Step 1: Write down the inequality

We start by writing down the given inequality: $$ 2x - 1 < x + 5 $$

Step 2: Subtract x from both sides of the inequality

To isolate x on one side and constants on the other side, we subtract x from both sides of the inequality: $$ 2x - x - 1 < x - x + 5 $$ This simplifies to: $$ x - 1 < 5 $$

Step 3: Add 1 to both sides of the inequality

To solve for x, we need to remove the constant term -1. To do this, we add 1 to both sides of the inequality: $$ x - 1 + 1 < 5 + 1 $$ This simplifies to: $$ x < 6 $$

Step 4: Write the solution

Now that we've isolated x, we can write the solution to the inequality: The solution to the inequality is \(x < 6\). This means that any value of x less than 6 will satisfy the given inequality.

Key Concepts

Solving Inequalities

Solving inequalities involves finding all possible values of a variable that make the inequality true. Just like equations, inequalities are mathematical statements with two sides. However, instead of an equal sign, inequalities use symbols such as <, >, ≤, or ≥ to show relationships. The main steps when solving an inequality include:

• Isolating the variable on one side.
• Performing similar operations on both sides of the inequality.

You need to remember that when you multiply or divide an inequality by a negative number, the inequality sign flips direction. This is a crucial difference between solving equations and inequalities. To better understand, consider starting with simpler numbers to experiment with inequality operations. Feel free to test boundary values to ensure that your solution maintains the inequality's truth.

Algebraic Manipulation

Algebraic manipulation entails systematically applying operations like addition, subtraction, multiplication, and division to both sides of an inequality to simplify or solve it. The goal is to isolate the variable, turning the inequality into a simpler form.In our example:

• We start with the inequality \(2x - 1 < x + 5\).
• The first step is algebraically manipulating the inequality by subtracting \(x\) from both sides, which simplifies to \(x - 1 < 5\). This is crucial to balance the inequality.
• Next, adding 1 to both sides provides the simple inequality \(x < 6\).

Algebraic manipulation is essential in solving inequalities, as it allows variables and constant terms to be paired down systematically until the inequality's solution becomes evident. It's these logical steps that ensure every manipulation keeps the inequality balanced.

Linear Inequality

A linear inequality is similar to a linear equation but uses inequality symbols instead. It represents a range of values rather than a single solution. This exercise demonstrated solving the linear inequality \(2x - 1 < x + 5\). Here’s how we approach linear inequalities:

• Recognize linear inequalities by their form, which resembles \(ax + b < c\) or \(ax + b > c\).
• Graphically, linear inequalities can be represented on a number line, showing a range that satisfies the inequality.
• They can also appear in graphs as a region defined by a line, where the line is not included in "<" or ">" inequalities (indicated by a dashed line) unless the inequality includes equality (≤ or ≥), which makes the line solid.

Linear inequalities are useful in various real-world situations where making sense of ranges is necessary, such as budgeting, understanding constraints, or forecasting outcomes. Understanding them equips us with the ability to interpret and solve practical problems.