Problem 54 - Elementary Algebra

Question

For the following problems, solve the inequalities. $$ 4 x+5>5 x-11 $$

Step-by-Step Solution

Answer

Answer: The values of x that satisfy the inequality are \(x < 16\) or in interval notation, \((-\infty, 16)\).

Step 1: Rearrange the inequality

To rearrange the inequality, subtract "4x" from both sides: $$ 4x + 5 - 4x > 5x - 11 - 4x $$ This simplifies the inequality to: $$ 5 > x - 11 $$

Step 2: Simplify the inequality

To simplify the inequality, add "11" to both sides: $$ 5 + 11 > x - 11 + 11 $$ This simplifies further to: $$ 16 > x $$

Step 3: Solve the inequality

Now, we can rewrite the inequality as $$ x < 16 $$ This inequality tells us that x must be less than 16.

Step 4: Write the solution in interval notation

We can express the solution using interval notation, which tells us the range of values for x that satisfy the inequality. In this case, the solution is: $$ (-\infty, 16) $$ So, the solution to the inequality \(4x + 5 > 5x - 11\) is \(x < 16\) or in interval notation, \((-\infty, 16)\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving inequalities and equations. It involves rearranging and simplifying mathematical expressions to uncover the desired solution. Here, we begin with the inequality \(4x + 5 > 5x - 11\). To solve it, you need to isolate the variable "\(x\)" on one side, which simplifies the problem.By subtracting \(4x\) from both sides, you eliminate \(4x\) from the left side. This is a strategic move because it simplifies the left-hand side to just a constant, leaving \(x\) on the right-hand side. This results in a simpler inequality: \(5 > x - 11\). This step is crucial because it reduces the complexity of the inequality, making it easier to solve. Simplifying expressions often involves such operations as subtracting or adding terms on both sides to maintain the inequality's truth.

Step-by-Step Solution

Solving inequalities through a step-by-step approach involves several straightforward stages, ensuring no important aspects are overlooked.

• **Step 1**: Rearrange the inequality to bring similar terms together on one side. For instance, by subtracting \(4x\) on both sides, we remove \(x\) terms from the left-hand side, leading to \(5 > x - 11\).
• **Step 2**: Simplify the inequality. In this exercise, add 11 to both sides to eliminate \(-11\) from the right-hand side. Doing so results in \(16 > x\).
• **Step 3**: Solve the simplified inequality. This can often involve switching the sides for clarity, hence it becomes \(x < 16\). This clearly indicates that \(x\) must be less than 16.

Every step in solving inequalities is about maintaining balance similar to a scale. Whatever you do to one side must be done to the other. This methodical approach ensures the inequality remains valid and the solution accurate.

Interval Notation

Interval notation is a compact method to express a range of values for which an inequality holds true. It is particularly useful for presenting solutions in a concise form. In this exercise, the inequality \(x < 16\) means "\(x\) is all numbers less than 16". In interval notation, this is represented as \((-\infty, 16)\).

• The left part of the interval, \(-\infty\), means there is no lower bound since \(x\) can take on indefinitely large negative values.
• The right part, \(16\), is the upper bound, and it is not included in the solution, which is why a parenthesis \((\) is used instead of a bracket \([\).

Interval notation provides a simple yet powerful way to give the solution of an inequality, making it easier to understand and communicate the range of solutions succinctly.