Question

Find the solution set of the conjunction \((1 / 2) \mathrm{x}+1>3\) and \(\mathrm{x}>2 \mathrm{x}-6\)

Short answer

The solution set of the conjunction \(\frac{1}{2}x+1 > 3\) and \(x > 2x-6\) is \(4 < x < 6\).

Step-by-step solution

Solve the first inequality

We have the first inequality: \(\frac{1}{2}x + 1 > 3\). Let's isolate x on one side: \[\frac{1}{2}x > 3 - 1\] \[\frac{1}{2}x > 2\] Now, multiply both sides of the inequality by \(2\): \[x > 4\] So, the solution set for the first inequality is \(x > 4\).

Solve the second inequality

We have the second inequality: \(x > 2x - 6\). Let's isolate x on one side: \[x - 2x > -6\] Combine like terms: \[-x > -6\] Now, multiply both sides of the inequality by \(-1\). Remember to reverse the inequality sign since we are multiplying by a negative number: \[x < 6\] So, the solution set for the second inequality is \(x < 6\).

Find the intersection of the solution sets

Now, we have two solution sets, \(x > 4\) and \(x < 6\). Since it's a conjunction, we need to find the intersection of these solution sets. The intersection of the solution sets will be the set of all x-values that satisfy both inequalities. The overlapping region of these two solution sets is \(4 < x < 6\). Thus, the solution set of the conjunction \(\frac{1}{2}x+1 > 3\) and \(x > 2x-6\) is \(4 < x < 6\).

Key concepts

Inequality Conjunctions

Understanding inequality conjunctions is essential when you are working with multiple inequalities that must all be satisfied at the same time. In our exercise, we have two separate inequalities to solve, and we must determine where their solutions overlap. This overlap is called the intersection of the sets. Think of it as the 'and' in everyday language—both conditions must be true.
For instance, if we're organizing a meeting that needs to fit into the busy schedules of two people, we look for a time slot that works for both. Similarly, solving inequalities in conjunction requires finding the values that make both inequalities true. It's important to remember that when working with inequality conjunctions, if there’s multiplication or division by a negative number, we must reverse the direction of the inequality sign.
To visualize the solution sets better, some students find it helpful to plot the inequalities on a number line or use a graph. This demonstrates the practical range and can clearly show the intersection, or the 'solution set,' where the x-values satisfy all parts of the conjunction. It provides a visual confirmation of the algebraic process we follow.

Solution Sets for Inequalities

The solution set of an inequality is the set of all possible values of the variable that make the inequality true. When we work on exercises with inequalities, the solution set can be represented in several ways including graphically, through interval notation, or by using inequality notation. For students, being familiar with these representations can be the key to a better understanding.
Of note is the fact that solution sets for inequalities can range from a single point to an infinite set of numbers. When represented on a number line, solutions to inequalities are typically indicated by a shaded region with open or closed dots at the boundaries to indicate whether the endpoints are included ('greater or equal to' / 'less or equal to') or not included ('greater than' / 'less than'). The interval notation conveys the same information in a compact form, using brackets and parentheses to express inclusion or exclusion of endpoints.
In our exercise, the solution set for the conjunction of the two inequalities is the range from 4 to 6, not including the endpoints. This is visually represented by a line segment between 4 and 6 on the number line with open circles at both 4 and 6. In interval notation, it is expressed as (4, 6).
Students should practice converting between these different representations of solution sets to become more comfortable with inequalities. Doing so can make it easier to understand the range of possible solutions and will help when encountering compound inequalities.

Linear Inequalities

Linear inequalities are similar to linear equations but with an inequality sign (>, <, ≥, or ≤) instead of an equals sign. The solutions to linear inequalities are not just single numbers, but a range of numbers that make the inequality true. These can be simple to solve and usually require the same initial steps as solving linear equations, such as simplifying expressions, combining like terms, and isolating the variable.
In the context of our exercise, we dealt with linear inequalities where we isolated the variable (x) on one side. By performing simple algebraic operations—addition, subtraction, multiplication, and division—we found the range of values for (x) that satisfied each inequality.

Why the Direction Matters

Remember that the direction of an inequality sign tells us the relationship between the variable and the rest of the expression. When graphing, this direction indicates which part of the number line to shade. In our example, solving the first inequality yielded (x > 4), which means we shade the region to the right of 4. The second inequality, after accounting for reversing the direction due to multiplying by -1, led us to shade to the left of 6. The graphical representation makes it clear why the intersection of these two shaded regions is the final solution.

Real World Relevance

Linear inequalities are not just academic exercises; they model real-world situations where there can be constraints with upper and lower bounds. An example could include budget limits or minimum and maximum temperatures for a chemical reaction. Learning to efficiently solve linear inequalities is a valuable skill that aids in problem-solving in various fields from economics to engineering.