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Chapter 2: Problem 54

Solve each compound inequality. Write solutions in interval notation. See Examples I through 8 . $$ 2 x-1 \geq 3 \text { and }-x>2 $$

Short Answer

Expert verified
The solution is an empty set: \(\emptyset\).

Step by step solution

01

Solve the first inequality

We need to solve the inequality \(2x - 1 \geq 3\). 1. Add 1 to both sides: \[2x - 1 + 1 \geq 3 + 1\] This simplifies to \(2x \geq 4\).2. Divide both sides by 2 to solve for \(x\): \[x \geq 2\]
02

Solve the second inequality

Now, solve the second inequality \(-x > 2\).1. Multiply both sides by -1 (remember to reverse the inequality when multiplying by a negative number): \[-1 \cdot (-x) < -1 \cdot 2 \] This simplifies to \(x < -2\).
03

Find the Intersection of the Solutions

The solutions to the compound inequality \(2x-1 \geq 3\) and \(-x > 2\) are \(x \geq 2\) and \(x < -2\) respectively.In this case, the two solutions do not overlap. Since there is no common \(x\) which satisfies both inequalities simultaneously, there is no solution to the compound inequality.
04

Write the Solution in Interval Notation

Since the solution sets for the inequalities \(x \geq 2\) and \(x < -2\) do not intersect, the solution in interval notation is the empty set. Thus, we write it as \(\emptyset\) or \((\)) for the answer.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interval Notation
Interval notation is a way of expressing a set of numbers or range of values. It provides a clear visual representation of where the solutions of an inequality lie on the number line. When writing in interval notation, we use brackets or parentheses to include or exclude endpoints of an interval.
  • Bracket [ or ]: Indicates that an endpoint is included (closed interval).
  • Parenthesis ( or ): Indicates that an endpoint is not included (open interval).
For example,
  • [2, 5] means all numbers between 2 and 5, including 2 and 5.
  • (2, 5) means all numbers greater than 2 and less than 5, excluding both 2 and 5.
  • (-\infty, +\infty) represents all real numbers.
In the case of compound inequalities like the one given, if there is no overlap of solutions, this is expressed as an empty set denoted by \(\emptyset\). This means that there is no number that satisfies both inequalities.
Solutions to Inequalities
Solutions to inequalities indicate the set of all possible values that can satisfy a particular inequality. To solve these inequalities, we must determine which values of the variable make the inequality statement true.
  • Linear inequalities: These involve variables raised only to the first power and no products of variables.
The solution to an inequality like \(2x - 1 \geq 3\) is found by isolating \(x\) on one side:
  • Add 1 to both sides to neutralize the \(-1\) on the left.
  • Divide the resultant inequality by 2 to solve for \(x\).
This process gives \(x \geq 2\). For the inequality \(-x > 2\), we multiply by -1, which reverses the inequality, and thus find \(x < -2\). Each of these results is a solution to one inequality, but they don't satisfy both simultaneously, leading to no overall solution in the compound context.
Solving Linear Inequalities
Solving linear inequalities involves finding the values for which a linear expression is greater or lesser than another value. The steps are similar to those for solving linear equations, except for the rule about reversing the inequality sign when multiplying or dividing by a negative number.In our example, we face two inequalities: \(2x - 1 \geq 3\) and \(-x > 2\).
  • Start by isolating the variable on one side of each inequality.
  • For \(2x - 1 \geq 3\): Add 1 to both sides to eliminate \(\-1\) and then divide by 2 to solve for \(x\).
  • For \(-x > 2\): Multiply both sides by -1. Remember to flip the inequality sign thus obtaining \(x < -2\).
It is crucial to graph or visualize these solutions to understand their implications properly. When solving compound inequalities, you need to combine the solutions, but if these solutions don't overlap, as in this exercise, there is unfortunately no common solution.

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