91Ó°ÊÓ

When encountering an algebraic inequality, it's quite like dealing with an equation. However, instead of finding a value that makes the equation true, we are searching for a range of values that satisfy the inequality. These types of problems ask us to determine the relative relationship between expressions.

In the given exercise, the process begins with expansion and simplification. By distributing the 4 across the \(x+3\), the algebra becomes clearer, transforming into a simpler inequality, \(4x + 12 \leq 44\). The next step to isolate the variable, \(x\), involves undoing any addition or subtraction around it. In this case, we subtracted 12 from both sides. Lastly, to further isolate \(x\), we divide by the coefficient (4), simplifying the inequality to \(x \leq 8\). This indicates that any number less than or equal to 8 makes the original inequality true.

To visualize the solution set of an inequality, graphing is an indispensable tool. A simple number line is sufficient to represent the solution to linear inequalities.

For the inequality \(x \leq 8\) that we simplified from the original exercise, the graph would showcase a closed circle at \(x=8\). This signifies that 8 is included in the solution set, corresponding to the 'less than or equal to' symbol. Shade the number line to the left of the circle to represent all numbers that are less than 8, which are also solutions to the inequality. Always remember, a closed circle is used for \(\leq\) or \(\geq\), while an open circle represents < or >, indicating that the exact value is not part of the solution.

A linear inequality is similar to a linear equation in that there's a linear expression involved. However, instead of equality, there's an inequality sign that must be respected throughout the computation process.

Handling linear inequalities like the one presented, \(4(x+3) \leq 44\), means treating the less-than-or-equal-to sign similarly to an equal sign, with a few additional rules to note: When multiplying or dividing by a negative number, the inequality sign flips. In our exercise, we avoided this complication since we divided by a positive number (4).

In applications, these inequalities are not just academic. They are used in a multitude of fields such as economics, engineering, and social sciences to describe constraints and conditions where there is a range of viable answers rather than just one.