Question

Solve the inequality. Graph the solution. \(2 x-3<5\)

Short answer

The solution to the inequality is \(x < 4\). This is represented by all numbers less than 4 on a number line.

Step-by-step solution

Isolate the variable x

The goal is to isolate x. Begin by adding 3 to both sides of the inequality to undo the subtraction. This gives \(2x < 5 + 3\), which simplifies to \(2x < 8\).

Divide by 2

Next, divide both sides of the inequality by 2 to isolate x. This yields the inequality \(x < 8/2\), which simplifies to \(x < 4\).

Graph the solution

On a number line, one would shade all the numbers less than 4 to represent the solution set and use a parenthesis (because 4 is not included in the solution).

Key concepts

Understanding Algebra

Algebra is all about using symbols, usually letters, to represent numbers and express mathematical relationships. It is a fascinating area of math because it allows us to solve problems even when we don't have all of the information. Algebra helps us translate real-world problems into easy-to-solve equations or inequalities.

In the given exercise, we worked with an inequality: \(2x - 3 < 5\). This inequality represents a condition that we need to fulfill, telling us the range of values that \(x\) can take. By manipulating the inequality using rules of algebra, similar to how we would manipulate an equation, we find the solution.

Remember, algebraic rules dictate that whatever operation you do to one side of the inequality, you must do to the other side as well. This keeps the inequality balanced at all times.

Solving Inequalities

Solving inequalities is a fundamental skill in algebra. An inequality expresses a relationship in which one quantity is greater than or less than another. Unlike equations, which are statements of equality, inequalities are statements of non-equality. Here are the steps involved in solving them:

• Isolate the Variable: Our first goal is to get the variable on one side of the inequality. In the step-by-step solution, we added 3 to both sides of the inequality \(2x - 3 < 5\) to remove the \(-3\) from the left side.
• Simplify: Perform any necessary arithmetic to simplify the inequality. We then divided the result by 2 on both sides to solve for \(x\), resulting in \(x < 4\).

A key thing to remember when solving inequalities is the rule for multiplying or dividing by a negative number. Unlike with equations, if you multiply or divide both sides by a negative number, you have to flip the inequality sign.

Graphing Inequalities

Graphing inequalities is a way to visually represent the solutions on a number line or coordinate plane. It helps us see which values satisfy the condition expressed by the inequality.

Let's talk about graphing on a number line, as in the original exercise. Here's how it works:

• Mark the Critical Value: Identify the critical number from the inequality. In our case, \(x < 4\), the critical value is 4.
• Shading: Shade the region on the number line that represents the solution set. For \(x < 4\), you would shade all the numbers to the left of 4.
• Boundary Note: Use a parenthesis at 4, suggesting \(4\) itself is not part of the solution.

Graphing makes it easy to comprehend the range of solutions and is a valuable visual aid in understanding how inequalities behave.