Question

Check whether the given number is a solution of the inequality. $$ 12 a \leq a-9 ;-2 $$

Short answer

-2 is a valid solution for the given inequality.

Step-by-step solution

Identify the Inequality

The inequality given in the problem is \(12a \leq a - 9\). The task is to check if -2 is a valid solution for this inequality.

Substitute the Value

Now, replace 'a' in the inequality with -2. This gives \(12(-2) \leq -2 - 9\). That simplifies to \(-24 \leq -11\).

Evaluate the Inequality

Evaluate which side is greater: -24 or -11. We know -11 is greater than -24. Hence, the inequality \(-24 \leq -11\) is true.

Key concepts

Algebraic Inequalities

Understanding algebraic inequalities is fundamental to mastering algebra. An inequality, unlike an equation, indicates that the values on one side are not necessarily equal to those on the other side. Instead, one side may be less than or greater than the other. Symbols such as \<\ (less than), \>\ (greater than), \leq\ (less than or equal to), and \geq\ (greater than or equal to) are used to denote these relationships. For example, \( 2 < 3 \) indicates that 2 is less than 3, and \( 5 \geq 5 \) confirms that 5 is greater than or equal to 5.

In the context of the given exercise, \( 12a \leq a - 9 \) expresses that the product of 12 times 'a' is less than or equal to 'a' minus 9. Understanding how to manipulate these inequalities is crucial in solving them, and often requires applying the same principles used in solving equations, such as performing inverse operations and keeping the inequality balanced. However, it is important to note that when multiplying or dividing both sides by a negative number, the direction of the inequality sign must be reversed to maintain the inequality's truth.

When checking if a number is a solution to an inequality, we look for the truthfulness of the statement. If substituting the number into the inequality makes the inequality valid, then the number is indeed a solution to the inequality.

Inequality Substitution

Inequality substitution is a key step in verifying whether a particular value satisfies an inequality. This method involves replacing the variable in the inequality with the number in question. It is analogous to plugging a value into a formula to get a result:if the result makes sense, then you've got the right input. In mathematical terms, substitution is the act of replacing a variable with its actual value.

Consider the given problem that asks us to check for \( a = -2 \) in the inequality \( 12a \leq a - 9 \). Step 2 from the provided solution shows this process in action: substitute \( -2 \) for every instance of 'a.' After substitution, the inequality becomes a comparison of numerical values, which can be much easier to evaluate.

Evaluating Inequalities

Evaluating inequalities is the final step in the problem-solving process. Once the substitution has been carried out, we compare the numerical expressions on each side of the inequality to determine their relationship. If the inequality holds true after the substitution, then the number is a solution.

In our exercise, after substituting -2 into the inequality, we get \( -24 \leq -11 \). Now we can evaluate: since -24 is indeed less than -11, the inequality stands correct, proving that -2 is a solution to the inequality. Understanding this concept helps in solving a wide range of problems beyond simple numerical checks; it's applicable in graphing solutions on a number line and solving for variable ranges in more complex algebraic expressions.

Remember, stay mindful of the rules of inequality when performing operations that may affect the inequality sign, such as multiplying or dividing by negative numbers. With practice, the process of solving inequalities can become a powerful tool in mathematics.