Problem 91 - Introductory Algebra for College Students

Question

Solve $$2 x+3<3(x-5)$$

Step-by-Step Solution

Answer

The solution to the inequality is \(x > 18\).

Step 1: Distribute on the Right Side

First, distribute the '3' on the right side of the inequality which gives: \(2x + 3 < 3x - 15\).

Step 2: Arrange Terms

Next, to isolate 'x', subtract '2x' from both sides which gives: \(3 < x - 15\).

Step 3: Simplify Further

Now, add '15' to both sides which gives the solution to the inequality: \(x > 18\). Finally, this means any value of 'x' that is greater than '18' satisfies the inequality.

Key Concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions that involve variables and show the relationship between two values that are not equal. Unlike equations, which state that two expressions are equal, inequalities tell us that one side is greater than, less than, greater than or equal to, or less than or equal to the other side.

For example, the inequality in our exercise, 2x + 3 < 3(x - 5), indicates that the value on the left-hand side, once the value of 'x' is substituted in, is less than the value on the right-hand side. Understanding how to manipulate and solve these inequalities is a fundamental skill in algebra and is applied extensively in various areas of mathematics and real-world problem-solving.

Linear Inequalities

Linear inequalities are similar to linear equations; however, instead of an equals sign, they have an inequality sign, such as <, >, \( \leq \), or \( \geq \). These inequalities graph as straight lines on a coordinate plane, but instead of just a line, a linear inequality represents a whole area on one side of the line.

In the exercise provided, we encounter a linear inequality. The given inequality 2x + 3 < 3(x - 5) will, once solved, indicate a range of values for 'x' that make the inequality true. It's important to note that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign should be reversed to maintain the correct relationship.

Inequality Solution Steps

To solve an inequality, we typically follow a series of steps similar to those used for solving a linear equation, with some important considerations specific to inequalities. Here are basic steps to solve simple inequalities:

• Distribute any terms within parentheses.
• Combine like terms and simplify both sides of the inequality.
• Isolate the variable by performing operations on both sides.
• Carefully consider the direction of the inequality if you multiply or divide by a negative number.
• Interpret the solution and write down the solution set.

For the exercise 2x + 3 < 3(x - 5), we followed similar steps. We began by distributing the '3' across the (x - 5) to eliminate parentheses. Next, we grouped the 'x' terms on one side by subtracting '2x' from both sides. Then we isolated 'x' by adding '15' to both sides, yielding the final solution x > 18. This solution tells us that the inequality holds for any 'x' value greater than '18'.