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When we talk about combining like terms, we refer to the process of simplifying expressions by adding or subtracting coefficients of the same variable. This is an essential first step in solving inequalities, and it helps make complex expressions more manageable.

In the original exercise, we start with the inequality: \[ 7x - 8 - 5x < 38 \]

Here, we have two terms with the variable \( x \), namely \( 7x \) and \( -5x \). To combine them, we simply subtract the coefficients: - From \( 7x \), take away \( 5x \), which gives us \( 2x \).
- The expression simplifies to \( 2x - 8 \), thus combining the like terms makes the inequality easier to solve.

This process helps simplify and reduce the length of the expression, making your next steps, such as isolation and solving, much more straightforward.

Once you've combined like terms, the next crucial step in solving an inequality is isolating the variable. This means getting the variable alone on one side of the equation so you can solve for its value. The goal is to make the inequality as clear and straightforward as possible.

After combining like terms, the expression becomes:\[ 2x - 8 < 38 \]

To isolate \( x \), you need to remove any additional numbers attached to it, which can be done using simple arithmetic:- Add 8 to both sides of the inequality to get: \[ 2x - 8 + 8 < 38 + 8 \]- This simplifies to: \[ 2x < 46 \]

Adding or subtracting the same number on both sides doesn't alter the inequality; it simply aids in balancing and isolating the variable. Once \( x \) is isolated, you're just one step away from finding its value.

With the variable isolated, we can proceed to solve the inequality. Solving linear inequalities involves similar steps to solving equations, but you should be mindful of certain key rules, especially when it comes to operations with negative numbers.

Looking at our example after isolating the variable:\[ 2x < 46 \]

The next step is to solve for \( x \) by dividing both sides of the inequality by the coefficient of \( x \), which is 2:- Perform the division: \[ \frac{2x}{2} < \frac{46}{2} \]- This results in: \[ x < 23 \]

Remember, when dividing or multiplying both sides of an inequality by a positive number, the inequality sign remains the same. If it were a negative number, you'd need to reverse the inequality sign. Once you perform this division, you have your solution for \( x \). Here, \( x \) simply needs to be less than 23 to satisfy the original inequality.