Question

Solve the inequality. $$-2 x<-14$$

Short answer

The solution to the inequality is \(x > 7\).

Step-by-step solution

Divide by -2

Divide each side of the inequality by -2 to isolate x. The rule with dividing or multiplying inequalities is to flip the inequality symbol when dividing or multiplying by a negative number. Thus, \(-2x / (-2) > -14 / (-2)\) which simplifies to \(x > 7\).

Interpret the solution

The solution \(x > 7\) means that any number greater than 7 is a solution to the inequality. So, for example, if you substitute 8 into the original equation, you would get \(-2*8 = -16\) which is indeed less than -14, and thus satisfies the inequality.

Key concepts

Inequality Symbol Rules

Understanding inequality symbol rules is essential for correctly solving and interpreting inequalities. When you encounter an inequality, such as \(-2x < -14\), it's crucial to remember that these are not the same as equations. Inequalities show a relationship of less than, greater than, less than or equal to, or greater than or equal to, rather than equality.

One of the fundamental rules is the direction of the symbol: '<' means 'less than', and '>' means 'greater than'. It is important to treat these symbols with care as they determine which values are part of the solution set. When we multiply or divide both sides of an inequality by a negative number, a pivotal concept comes into play: we must flip the direction of the inequality symbol. This is not something we do when dealing with equations.

Let's consider the provided textbook problem \(-2x < -14\). When we divide both sides by -2 to isolate the variable x, we flip the inequality symbol, changing '<' to '>', leading to the solution \(x > 7\). This rule maintains the truth of the original inequality across the manipulation of terms.

Dividing by a Negative Number

Dividing by a negative number can cause confusion, but it's an essential concept in solving inequalities correctly. The golden rule to remember is: whenever you divide or multiply both sides of an inequality by a negative number, the inequality symbol must be reversed.

This counterintuitive step is important because negative numbers have the property of reversing order when combined with inequalities. Imagine the number line: when we multiply or divide by a negative number, the direction of the order on this line flips. Therefore, the inequality must flip to preserve the original relational condition between the variables.

In the exercise \(-2x < -14\), we divide by '-2', thus the '<' becomes '>'. Neglecting to flip the symbol can lead to an entirely incorrect solution set, so this rule is crucial for success. Always be vigilant for this step when you're working through inequalities involving negative operations.

Interpreting Inequality Solutions

Once an inequality is solved, it's critical to correctly interpret the solution. The inequality \(x > 7\), for instance, tells us that x can be any number greater than 7. This is not just a single number, but an entire range of numbers to infinity.

Understanding this concept allows us to realize that inequalities often have infinite solutions. To visualize this, you might think of a number line: every point on the line that lies to the right of 7 is included in the solution. This interpretation helps in graphing the solution and applying it to real-life scenarios.

Also, it's important to check the validity of your solution by picking numbers within your solution set. As demonstrated in the exercise, substituting a number greater than 7, such as 8, into the original inequality confirms that it holds true. This verification step is a good habit to ensure that the solution is correctly understood and applied.