Problem 16 - Elementary Algebra

Question

Solve the equations and inequalities for the following problems. $$ \frac{-4 x-3}{3}>-9 $$

Step-by-Step Solution

Answer

Answer: The range of values for x is $(-\infty, 6)$.

Step 1: Eliminate the Fraction

To eliminate the fraction, multiply both sides of the inequality by the denominator, which is 3. $$ \frac{-4x - 3}{3} \times 3 > -9 \times 3 $$ This simplifies the inequality to: $$ -4x - 3 > -27 $$

Step 2: Isolate the Variable

Next, we'll isolate the variable x by adding 3 to both sides of the inequality: $$ -4x > -24 $$ Now, we'll divide both sides by -4 to solve for x. Remember that when we divide or multiply both sides of an inequality by a negative number, the inequality symbol flips: $$ x < \frac{-24}{-4} $$

Step 3: Simplify and Write the Solution

Simplify the inequality: $$ x < 6 $$ The solution to this inequality is: $$ x \in (-\infty, 6) $$

Key Concepts

Algebraic Inequalities

Algebraic inequalities are mathematical expressions that show the relationship between two values in which one is not exactly equal to the other, but rather less than or greater than. Unlike equations, where the goal is to find an exact value, inequalities are used to determine a range of possible values that satisfy the condition put forth. This could be an invaluable tool when dealing with real-life situations like budgeting where exact numbers may not be available, but ranges can be estimated.

For example, if you want to understand how much you can spend on a meal without going over budget, an inequality would help to express the maximum amount you could spend. Understanding how to solve algebraic inequalities helps students interpret and solve problems where precision is not the key, but a spectrum of possibilities is acceptable.

Isolate the Variable

When solving inequalities, 'isolating the variable' means rearranging the inequality so that the variable we are solving for is by itself on one side. This is similar to solving equations. We perform operations such as adding, subtracting, multiplying, and dividing both sides of the inequality to achieve this isolation.

It is pivotal that while performing these operations, the inequality's balance is maintained. Any operation done to one side must also be done to the other side. This principle ensures that the inequality remains true and the solution is valid. When the variable is isolated, we get a clear picture of its possible values in relation to the other side of the inequality.

Eliminating Fractions in Inequalities

Fractions can make inequalities appear more complex than they are. To simplify these inequalities, we can eliminate fractions by finding the least common denominator or by simply multiplying both sides of the inequality by the denominator of the fraction involved.

Doing so effectively 'clears' the fraction, simplifying the inequality to a form that is usually easier to solve. In our example, multiplying by 3, the denominator of \(\frac{-4x - 3}{3}\), we eliminated the fraction leading to a simpler inequality of \-4x - 3 > -27\. This step is essential because it helps prevent any mistakes that could arise from working directly with fractions in inequalities.

Inequality Symbol Reversal

A critical rule when working with inequalities is the 'inequality symbol reversal'. This occurs when you multiply or divide both sides of an inequality by a negative number. As this fundamentally changes the order of the values, the inequality symbol must be flipped to maintain the true relationship between them.

For instance, if you have an inequality like \(-2x > 8\), dividing both sides by \(-2\) to isolate \(x\) flips the 'greater than' symbol to a 'less than' symbol, so the inequality then reads \(x < -4\). Remembering this rule will ensure that the solutions to inequalities remain accurate after performing such operations.