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In algebra, inequalities are a way to express a relationship where one side is not necessarily equal to the other, but rather greater than, less than, or within a certain range compared to another. The most common symbols used in algebraic inequalities are less than (<), greater than (>), less than or equal to (≤), and greater than or equal to (≥).

When dealing with algebraic inequalities, we follow similar procedures as with regular algebraic equations, with the exception of a critical rule regarding the multiplication or division by a negative number, which requires us to reverse the inequality sign.

• If a b, and you add or subtract the same positive number from both a and b, the relationship remains the same.
• If a b, and you multiply or divide both a and b by a positive number, the relationship doesn't change.
• If you multiply or divide both a and b by a negative number, the inequality sign reverses (a becomes >, and vice versa).

Manipulating inequalities strategically can help reveal the solution to an algebraic problem. Similar to solving equations, manipulation of inequalities involves adding, subtracting, multiplying, or dividing both sides of the inequality by the same number to maintain the truth of the original statement.

How you manipulate an inequality can impact the ease with which you find a solution. For instance, in the given exercise, adding 14 to both sides of the given inequality eliminates the negative term and simplifies the problem, moving you one step closer to finding the range of values for x.

Essential Points to Note:

• Always perform the same operation on both sides of the inequality to maintain balance.
• Be aware of the multiplication or division by negative numbers. This calls for the flip of the inequality sign.
• Checking your solution by substituting a value within the range back into the original inequality can confirm its accuracy.

Linear inequalities are similar to linear equations but instead of an equals sign, an inequality symbol is used. Solving linear inequalities, like the one in our exercise, involves finding a set of all possible values that satisfy the inequality. Linear inequalities can have one variable (as in our exercise) or two or more variables.

For a one-variable linear inequality, the solution is usually a range of numbers. Strategies for solving simple one-variable linear inequalities include:

• Isolating the variable on one side of the inequality.
• Understanding that multiplying or dividing by a positive number does not affect the inequality.
• Graphical representation of the solution can help visualize the range of the solution on a number line.

Expressing the solution to a linear inequality can also involve interval notation or inequality notation, whichever is more suitable for the context. The provided example, where x must be less than 3, can be written as (-∞, 3) in interval notation. This notation suggests that x can be any number less than 3, not including 3 itself.