Algebra Study Guide

Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations which assert equality, inequalities describe a range of possible values for a variable. Solving an inequality means finding all the values of the variable that make the inequality true. The methods for solving linear inequalities are very similar to those for solving linear equations, but with one crucial difference: if you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality sign.

For example, to solve $2x - 3 < 7$:

1. Add 3 to both sides: $2x - 3 + 3 < 7 + 3 \implies 2x < 10$.
2. Divide both sides by 2: $\frac{2x}{2} < \frac{10}{2} \implies x < 5$. The solution is all numbers less than 5.

Consider solving $-3x + 1 \ge 10$:

1. Subtract 1 from both sides: $-3x + 1 - 1 \ge 10 - 1 \implies -3x \ge 9$.
2. Divide both sides by -3. Since we are dividing by a negative number, we must reverse the inequality sign: $\frac{-3x}{-3} \le \frac{9}{-3} \implies x \le -3$. The solution is all numbers less than or equal to -3.

It's important to remember this rule regarding negative multiplication/division to avoid errors. Inequalities can also be represented on a number line, using an open circle for < or > and a closed circle for ≤ or ≥ to indicate whether the endpoint is included in the solution set.

Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). Unlike equations which state that two expressions are equal, inequalities show a range of possible values for a variable. In one variable, this means the solution can be represented on a number line.

Solving inequalities is very similar to solving equations. The goal is to isolate the variable on one side of the inequality symbol. You can add or subtract the same number from both sides, and you can multiply or divide both sides by the same positive number. However, there is one crucial rule: if you multiply or divide both sides of an inequality by a negative number, you must reverse the direction of the inequality symbol. For example, if you have -2x < 6, dividing by -2 changes it to x > -3.

After solving an inequality, it's often helpful to represent the solution on a number line. A hollow circle is used for strict inequalities (< or >) to indicate that the endpoint is not included in the solution. A solid (filled) circle is used for inclusive inequalities (≤ or ≥) to indicate that the endpoint is included. An arrow extending from the circle shows the direction of the values that satisfy the inequality.

For example, if you solve an inequality and get x > 3, you would draw a number line, place a hollow circle at 3, and draw an arrow pointing to the right (towards numbers greater than 3). If you get x ≤ -1, you would draw a number line, place a solid circle at -1, and draw an arrow pointing to the left (towards numbers less than or equal to -1).

Understanding inequalities is fundamental to representing real-world constraints and conditions, such as limits on speed, capacity, or budget.