91Ó°ÊÓ

Simplifying inequalities makes solving them much easier. The main goal is to get the variable alone on one side of the inequality.
For example, let's start with the inequality from the exercise: \(2x < 3x + 7\).
The first step is to remove any terms that are making it harder to isolate the variable. Here, you can see the term \(3x\) on the right side.
By subtracting \(3x\) from both sides, we can simplify this to get \(2x - 3x < 7\).
The same approach is useful in the second inequality \(21x - 19 \geq 4x + 15\).
By subtracting \(4x\) from both sides, it becomes \(21x - 4x - 19 \geq 15\).
These steps help peel away the complexity and make it easier to work with the inequality.

Combining like terms is when we add or subtract terms that have the same variable or are constants. This is important for simplifying expressions.
In our exercise, once we subtract \(3x\) from both sides of \(2x < 3x + 7\), we get \(2x - 3x < 7\). Now, \(2x - 3x\) are like terms as they both have the variable \(x\).
Combining them, we get \(-x < 7\).
Similarly, for \(21x - 19 \geq 4x + 15\), after subtracting \(4x\), we have \(21x - 4x - 19 \geq 15\).
By combining the \('x'\) terms, it simplifies to \(17x - 19 \geq 15\).
Ensure always to perform these operations correctly to keep the inequality balanced.

While solving inequalities, you may need to reverse the inequality sign. This is a crucial step when multiplying or dividing both sides by a negative number.
For instance, in the step \(-x < 7\), to solve for \('x'\), divide both sides by \(-1\): \(\frac{-x}{-1} \geq \frac{7}{-1}\).
Note that dividing by \(-1\) reverses the inequality sign, giving us the solution \(x > -7\).
It is important to remember not to miss this step. Reversing the inequality sign ensures your solution is accurate and preserves the inequality's meaning.

Isolating the variable means getting the variable alone on one side of the inequality.
For the first inequality \(-x < 7\), it's straightforward once we deal with the negative coefficient, ending with \(x > -7\).
For the second inequality \(17x - 19 \geq 15\), we isolate \('x'\) by adding \(19\) to both sides, resulting in \(17x \geq 34\).
Finally, we divide by \(17\): \(\frac{17x}{17} \geq \frac{34}{17}\), which simplifies to \(x \geq 2\).
Consistently following these steps will help achieve a clear solution for any inequality.