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When solving inequalities, one of the first things you typically need to do is isolate the variable. This means getting the variable you are solving for by itself on one side of the inequality. In this example, we start with the inequality To isolate the variable term, you need to remove other terms from the side of the inequality where the variable is located. Here, you can subtract 4 from both sides of the inequality: \(-x + 4 - 4 > 3 - 4 \) By subtracting 4 on both sides, we get \(-x > -1 \). Remember, whatever mathematical operation you do to one side of the inequality, you must do to the other side as well. This keeps the inequality balanced. Isolating the variable is a crucial step because it simplifies the inequality, making it easier to solve.

Sometimes, when solving an inequality, you might need to multiply or divide by a negative number. The key rule here is that whenever you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. In our example, we have the simplified inequality \(-x > -1 \). To solve this, we need to get rid of the negative sign in front of the variable. To do so, multiply both sides by \(-1 \): \((-1)(-x) < (-1)(-1) \) Notice that we reversed the inequality sign from '>' to '<'. This is because multiplying by a negative changes the direction of the inequality. Therefore, our inequality becomes: \(x < 1 \). Always remember this crucial step, as forgetting to reverse the sign can lead to incorrect solutions.

Solving inequalities is an essential part of intermediate algebra that involves understanding how to manipulate algebraic expressions while maintaining the inequality relationships. It builds on basic algebra concepts like addition, subtraction, multiplication, and division, but adds more rules like the one about reversing inequality signs. Here are some general tips when solving inequalities:

• Always perform the same operation on both sides to keep the inequality balanced.
• Take special care when multiplying or dividing by negative numbers; remember to reverse the inequality sign.
• Double-check your final solution to make sure it satisfies the original inequality.

With intermediate algebra, you will see a mixture of different types of problems, including linear equations and quadratic inequalities. Being comfortable with these foundational concepts will help you tackle more complex algebraic problems in the future.