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Chapter 12: Problem 253

Solve the inequality \(2 \mathrm{x}-5>3\).

Short Answer

Expert verified
The solution to the given inequality \(2x - 5 > 3\) is \(x > 4\) or \((4,\infty)\) in interval notation.

Step by step solution

01

Add or Subtract to Isolate x-term

We need to isolate the term with the variable x (i.e., \(2x\)), so let's add 5 to both sides of the inequality to eliminate the constant term: \[2x - 5 + 5 > 3 + 5\] Simplifying, we get: \[2x > 8\]
02

Multiply or Divide to Isolate x

Now we need to isolate x further by dividing both sides of the inequality by 2: \[\frac{2x}{2} > \frac{8}{2}\] Simplifying, we get: \[x > 4\]
03

Write the Solution

The solution to the inequality is \(x > 4\). In interval notation, this can be written as: \((4,\infty)\) So the solution to the given inequality \(2x - 5 > 3\) is \(x > 4\) or \((4,\infty)\) in interval notation.

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Key Concepts

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

Inequality Solving Steps
Solving inequalities is similar to solving equations. The goal is to isolate the variable, and you can use algebraic operations like addition, subtraction, multiplication, or division to achieve this. Here鈥檚 a simple breakdown of the process:
  • Step 1: Add or Subtract to Isolate x-term
    Start by moving constants to the other side of the inequality. This often involves adding or subtracting terms. For example, in \(2x - 5 > 3\), add 5 to both sides to get \(2x > 8\).

  • Step 2: Multiply or Divide to Isolate x
    Next, you need to isolate \(x\) by dividing or multiplying, just like you would in an equation. In the example, divide both sides by 2, resulting in \(x > 4\).

  • Step 3: Write the Solution
    Once \(x\) is isolated, you can express the solution in various forms. Here, it's written as \(x > 4\).

Unlike equations, be careful when multiplying or dividing by a negative number, as this requires flipping the inequality sign. This rule is crucial to remember.
Interval Notation
When expressing the solution set of an inequality, interval notation is a concise and visually intuitive method. It describes the range of values that satisfy the inequality.
  • Understanding Interval Brackets
    An open interval, using parentheses, \((a, b)\), includes all numbers between \(a\) and \(b\), but not \(a\) or \(b\) themselves. A closed interval, using brackets, \([a, b]\), includes \(a\) and \(b\) as well.

  • Using Infinity
    Infinity symbols \((\infty\) and \(-\infty\)) are always paired with open parentheses because infinity is not a number that can be reached. For the inequality \(x > 4\), the interval notation is \((4, \infty)\), which means all values greater than 4.

This notation efficiently communicates the solution set without ambiguity, making it a preferred method in mathematics.
Algebraic Manipulation
Algebraic manipulation is the process of moving terms around within an inequality or equation to isolate variables. This skill is foundational in algebra and crucial for solving inequalities effectively.
  • Transposing Terms
    Transposing means moving a term from one side of an inequality to the other by adding or subtracting it. This step helps in simplifying the inequality, as seen when moving \(-5\) from \(2x - 5 > 3 \) to \(2x > 8\).

  • Clearing Fractions and Coefficients
    Once constants are transposed, clear out coefficients by using division or multiplication. In our example, dividing both sides by 2 isolates \(x\), making \(x > 4\).

Mastering algebraic manipulation involves practicing these operations until they become intuitive, paving a smoother path to finding solutions in both inequalities and complex algebraic equations.

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Most popular questions from this chapter

Solve \(1 / 6 \mathrm{x}-3<3 / 4 \mathrm{x}+1 / 2\)

Solve \(2 \mathrm{x}-3 \mathrm{y} \geq 6\)

Find the solution set of the disjunction \(2-3 x>5\) or \(2 x-1>5\)

Replace the inequality \(1<\mathrm{x}<3\) by a single inequality involving an absolute value.

Find the solution set of \(|2 \mathrm{x}+5| \leq \mathrm{x}+3\).
See all solutions

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