91Ó°ÊÓ

When working with inequalities, like with equations, the first step often involves simplifying expressions by combining like terms. Like terms are terms that have the same variables raised to the same power. For instance, in the inequality \(-x - 4 - 3x > 5\), the terms \(-x\) and \(-3x\) are like terms. They both contain the variable \(x\) raised to the first power.

To combine them, you simply add their coefficients. The coefficient of \(-x\) is \(-1\) and for \(-3x\), it is \(-3\). Adding these together gives you \(-1 - 3 = -4\). So, \(-x - 3x\) simplifies to \(-4x\).

Combining like terms helps to condense the inequality, making it easier to solve. Once you've simplified it, you're left with a cleaner inequality: \(-4x - 4 > 5\). This simplification step sets the stage for efficiently isolating the variable.

The goal of solving an inequality is to find the range of values that satisfy it. To do this, you need to isolate the variable on one side of the inequality. In our example, the inequality after simplifying is \(-4x - 4 > 5\). The next step is to move the constant term so that the variable \(x\) is isolated.

To eliminate the constant \(-4\) from the left side, add \(4\) to both sides of the inequality:

• Expression on the left: \(-4x - 4 + 4\)
• Expression on the right: \(5 + 4\)

This leads to \(-4x > 9\).

Remember, isolating the variable is about shifting terms across the inequality without breaking the equality. It brings you closer to knowing the potential values \(x\) can assume that satisfy the original inequality. Once we have \(-4x > 9\), the variable \(x\) is almost isolated, requiring just one more operation.

The final part of solving this inequality involves both division and an important rule regarding inequalities. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign. This rule is crucial to remember to maintain the inequality's truth.

In our example, we have \(-4x > 9\). The variable \(x\) is still not isolated because it is multiplied by \(-4\). To isolate \(x\), divide both sides by \(-4\):

• Left side: \(-4x/-4 = x\)
• Right side: \(9/-4 = -\frac{9}{4}\)

This gives \(x < -\frac{9}{4}\).

Notice that the inequality sign '>' became '<'. This is because dividing by a negative number effectively flips the inequality. The solution \(x < -\frac{9}{4}\) or \(x < -2.25\) tells you all the values that \(x\) can take to satisfy the original inequality. Ensuring proper reversal of inequality is key for accurate solutions.