Problem 86 - Precalculus

Question

In Exercises 85 and \(86,\) for the system of linear inequalities, assume \(a, b, c,\) and \(d\) are real numbers. $$\begin{array}{l}x \geq a \\\x < b \\\y > c \\\y \leq d\end{array}$$ What will the solution be if \(a>b\) and \(c>d ?\)

Step-by-Step Solution

Answer

No solution exists for this system of inequalities.

Step 1: Analyze the Inequalities

We have four inequalities involving variables \(x\) and \(y\): \(x \geq a\), \(x < b\), \(y > c\), and \(y \leq d\). These inequalities define the possible values \(x\) and \(y\) can take.

Step 2: Determine the Condition x-values

The inequalities \(x \geq a\) and \(x < b\) suggest \(x\) must be greater than or equal to \(a\) but less than \(b\). This implies \(x\) is in the interval \([a, b)\).

Step 3: Assess Given Condition for x

If \(a > b\), then the interval \([a, b)\) is invalid, as there are no real numbers that are simultaneously greater than or equal to \(a\) and less than \(b\). Hence, no real \(x\) satisfies both inequalities.

Step 4: Determine the Condition y-values

The inequalities \(y > c\) and \(y \leq d\) suggest \(y\) is in the interval \((c, d]\).

Step 5: Assess Given Condition for y

If \(c > d\), then the interval \((c, d]\) is invalid, as there are no real numbers that are simultaneously greater than \(c\) and less than or equal to \(d\). Hence, no real \(y\) satisfies both inequalities.

Step 6: Conclusion from Conditions

With the given conditions \(a > b\) and \(c > d\), both sets of inequalities for \(x\) and \(y\) are unsatisfiable. Thus, there is no solution that satisfies all the inequalities in the system.

Key Concepts

Real Numbers

The concept of real numbers is fundamental in understanding systems of inequalities. Real numbers include all the numbers on the number line, encompassing both rational numbers (like fractions and integers) and irrational numbers (numbers that cannot be expressed as simple fractions, like \(\sqrt{2}\) or \(\pi\)). Real numbers can be small, large, positive, negative, or even zero, giving us a wide range of possibilities when solving inequalities.
When dealing with inequalities involving real numbers, we aim to find solutions that satisfy all conditions specified by the inequalities in a system.
For instance, if you have an inequality like \(x \geq a\), it implies you are looking for real number solutions for \(x\) that start from \(a\) onwards on the number line. Similarly, \(y > c\) would indicate that the acceptable real values for \(y\) start from just above \(c\). Understanding the broad and inclusive nature of real numbers is crucial in pinpointing solution ranges in inequalities.

Interval Notation

Interval notation is a useful way to express ranges of real numbers that satisfy given inequalities. With interval notation, we use the symbols \([\) and \(]\) to indicate inclusive boundaries and \((\) and \()\) to denote exclusive boundaries. For example:

• \([a, b]\) represents all real numbers between \(a\) and \(b\), including the endpoints \(a\) and \(b\).
• \((a, b)\) denotes all real numbers between \(a\) and \(b\) but excludes the endpoints.
• \([a, b)\) indicates that the range includes \(a\) but not \(b\).
• \((a, b]\) includes \(b\) but not \(a\).

In the context of inequalities, interval notation efficiently communicates the set of solutions. For instance, if given \(x \geq a\) and \(x < b\), we would express the solution with interval notation as \([a, b)\). If no real number satisfies the condition, typically because the inequality interval is impossible (e.g., \([c, d)\) when \(c > d\)), we denote it as an empty set because no numbers can meet the criteria.

Solution of Inequalities

Solving inequalities involves finding the set of all real numbers that satisfy one or several conditions expressed by the inequalities. To effectively solve a system of linear inequalities, you need to consider each inequality and the relationships between them one at a time.
In our given problem, when two conditions like \(a > b\) and \(c > d\) exist, you quickly determine there are no solutions. This is because such conditions create logical contradictions; it's impossible for a real number to be greater than \(a\) and less than \(b\) simultaneously if \(a\) is greater than \(b\). Similarly, no \(y\) value can satisfy \(y > c\) and \(y \leq d\) if \(c\) is greater than \(d\).
When attempting to solve systems of inequalities, always check each inequality's conditions and their implications on potential solutions. Clearly identifying and understanding these logical relations help in determining whether a solution exists and, if so, what the solution set is.