91Ó°ÊÓ

Algebraic inequalities are mathematical expressions involving unknown variables that do not indicate exact equality (e.g. x + 3 > 2) but rather specify a range of possible values that will satisfy the inequality. To solve an inequality algebraically, like the one in our exercise \( -x + 6 \leq 2x + 9 \), the primary goal is to isolate the variable on one side of the inequality. This is achieved by conducting similar operations as one would do in an equation: adding, subtracting, multiplying, or dividing both sides by the same non-zero number.

However, there's a special rule to remember: when you multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign changes. For example, if you were to divide \( -2x > 4 \) by \( -2 \) to isolate \( x \), you'd invert the \( > \) to \( < \) so it would become \( x < -2 \). This rule is essential for correctly solving inequalities. After isolating the variable and solving the inequality as shown in the step-by-step solution, you can clearly see the range of possible values for \( x \).

Representing inequalities on a number line gives us a visual way to interpret the range of solutions. In our example, starting with \( -x + 6 \leq 2x + 9 \), we eventually find that \( x \) must be greater than or equal to -1. On a number line, this translates to a filled dot at -1 because the inequality includes equals to (\leq). The filled dot at -1 demonstrates that -1 is a part of our solution set.

From that point, a line extends to the right, representing all numbers greater than -1 that also satisfy the inequality. If the inequality did not include equality (\lt or \gt), we would depict this on the graph with an open dot. The graphical representation is a crucial step in not only showing the solution but also ensuring that the understanding of the inequality’s implications is accurate and visual.

Interval notation is a succinct way to write the set of all possible solutions to an inequality. It uses brackets and parentheses to indicate whether endpoints are included or not. In our exercise, the interval notation for the solution of \( -x + 6 \leq 2x + 9 \) is [ -1, \(\infty\))]. A closed bracket, [ or ], means that the endpoint number is part of the solution, as it is the case for -1 in our solution. On the other hand, an open parenthesis, ( or ), indicates that the endpoint is not included – which is how we represent the idea of infinity, since we cannot reach infinity itself.

Interval notation is especially useful for expressing solutions that include a range of numbers since it presents the information clearly and compactly. Remember, intervals may be finite, like [3, 8), or infinite, like the one in our exercise, showing one end stretches towards infinity. This notation is essential for communicating solutions in higher mathematics effectively.