91Ó°ÊÓ

When solving inequalities, the goal is similar to solving equations: to find the values of the variable that make the inequality true. However, unlike equations, inequalities indicate a range of possible solutions rather than a single value. In our example:\[ \frac{3x}{10} + 1 \geq \frac{1}{5} - \frac{x}{10} \]we aim to isolate the variable, in this case, \(x\). The inequality sign offers direction, showing whether values are larger, smaller, or both. It is important to remember that when multiplying or dividing both sides of an inequality by a negative number, the inequality sign flips direction. This key difference from equations can change the entire solution, so careful handling is crucial. In our example, solving the inequality involved moving all \(x\)-terms to one side and constants to the other; a typical first step that treats positivity straightforward without any sign changes due to negative multiplications or divisions.

Combining like terms is a fundamental step in solving algebraic expressions or inequalities. It involves grouping similar terms to simplify the process. Terms that contain the same variable raised to the same power are like terms and can be combined by adding or subtracting their coefficients. In the inequality given:\[ \frac{3x}{10} + 1 \geq \frac{1}{5} - \frac{x}{10} \]we want to consolidate the \(x\)-terms on one side. This involves moving \(-\frac{x}{10}\) to the left side to join with \(\frac{3x}{10}\). Doing so results in the following expression:\[ \frac{3x}{10} + \frac{x}{10} \geq \frac{1}{5} - 1 \]Now we can combine them as \( \frac{3x}{10} + \frac{x}{10} = \frac{4x}{10} \). This consolidation has simplified the inequality, making continuation to the next steps more straightforward.

Simplifying algebraic expressions means reducing them to a form that is easier to manage, often by eliminating fractions or combining terms. In the context of our inequality, after combining like terms, we reached:\[ \frac{4x}{10} \geq -\frac{4}{5} \]Here, simplifying involves getting \(x\) by itself. This is done by removing the fraction with \(x\) by multiplying both sides by the denominator, 10, which neatly cancels it:\[ 4x \geq -8 \]Finally, divide both sides by 4 to solve for \(x\):\[ x \geq -2 \]This step shows the critical nature of simplification in solving inequalities. By reducing expressions to their simplest form, the path to finding an answer becomes clearer and less prone to complication or error.