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Chapter 3: Problem 36

Solve each inequality. $$2 x+5<3 x+8$$

Short Answer

Expert verified
\(x > -3\)

Step by step solution

01

- Subtract 2x from both sides

To isolate the variable on one side, subtract 2x from both sides of the inequality:\[2x + 5 - 2x < 3x + 8 - 2x\]This simplifies to:\[5 < x + 8\]
02

- Subtract 8 from both sides

To isolate the variable, subtract 8 from both sides of the inequality:\[5 - 8 < x + 8 - 8\]This simplifies to:\[-3 < x\]
03

- Rewrite the inequality

The inequality \(-3 < x\) can be rewritten in a more standard form:\[x > -3\]This is the solution to the inequality.

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Key Concepts

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

Inequality Simplification
When solving inequalities, the first step involves simplifying the inequality to make it easier to work with. Simplification means getting rid of any unnecessary terms to reveal the core structure of the inequality. In our exercise, we start with the inequality \(2x + 5 < 3x + 8\)To simplify this, we remove the same terms from both sides of the inequality to avoid changing its meaning.
Variable Isolation
Isolating the variable involves getting the variable (often \(x\)) by itself on one side of the inequality. This isolates it from the constants (numbers without variables). In our exercise, we perform the following steps:
  • Subtract \(2x\) from both sides: \[2x + 5 - 2x < 3x + 8 - 2x\]
  • Simplify to: \[5 < x + 8\]
  • Subtract \(8\) from both sides: \[5 - 8 < x + 8 - 8\]
  • Simplify to: \[-3 < x\]
This series of operations isolates \(x\), making the inequality easier to solve.
Elementary Algebra
Elementary algebra involves basic algebraic operations needed to solve equations and inequalities. In our exercise, we use subtraction to manipulate the inequality step-by-step:

1. We start with the given inequality and simplify: \[2x + 5 < 3x + 8\]2. Subtract \(2x\) from both sides to target the variable:\[5 < x + 8\]3. Subtract \(8\) from both sides to further isolate \(x\): \[-3 < x\]In elementary algebra, it鈥檚 crucial to perform the same operation on both sides to maintain the balance of the inequality. The final solution is often rewritten in a more familiar form: \[x > -3\]. Understanding these basic steps is essential for solving more complex algebraic problems.

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