91Ó°ÊÓ

When we talk about integer values, we're looking at whole numbers that can be positive, negative, or zero, but without any fractions or decimals. An important aspect of solving inequalities is to identify which integers satisfy the given conditions.

In the exercise provided, the challenge is to find the integers that are greater than -6 and simultaneously less than -2. It's pretty much like looking for the numbers that are sitting between -6 and -2 on the number line. The integers that lie snugly in this range without touching the boundaries are -5, -4, and -3. These are the only integers that make the statement \( -6 < x < -2 \) true. It is crucial to leave out the endpoints, -6 and -2, because the inequality is strict—with the < signs indicating 'strictly greater than' and 'strictly less than'.

The process of solving inequalities might seem daunting at first, but it's similar to solving equations, with a key difference in how we treat the inequality when multiplying or dividing by a negative number. When confronted with an inequality, such as the one in our exercise \( -6 < x < -2 \) we're looking for a set of numbers that make the inequality true, rather than a single solution like in an equation.

It's helpful to visualize the number line and shade in the region that satisfies the inequality. Remember to use open circles to indicate that the end values aren't included if the inequality doesn't have an 'equals to' part, like \( \leq \) or \( \geq \) symbols. In this case, integers are distinct points on the number line and picking the correct ones within the given range is critical. Be sure to consider the direction of the inequality, as flipping an inequality sign can also flip your solution set.

An algebraic expression is a combination of variables, numbers, and operators. For example, \( 2x - 5 \) is an algebraic expression with \( x \) as the variable. When we're solving inequalities that involve algebraic expressions, we apply similar operations as we do with arithmetic expressions—but we keep an eye on the direction of the inequality sign.

If the inequality involves a simple algebraic expression without the need to conduct operations on both sides, such as the expression given in our exercise \( -6 < x < -2 \) the solution involves understanding the relationship presented by the inequality and determining which values for the variable \( x \) maintain the truth of the entire expression. Working with algebraic expressions requires careful manipulation; one must be mindful of each step, especially when distributing or combining like terms.