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Chapter 10: Problem 145

Solve: \(3 x-4 \leq 2\) and \(4 x+5 \geq 5\) (Section 9.2, Example 2)

Short Answer

Expert verified
The solution is \(0 \leq x \leq 2\).

Step by step solution

01

Solve the first inequality

The first inequality is \(3 x-4 \leq 2\). To solve for x, firstly, isolate the term with x by adding 4 to both sides of the inequality: \(3x \leq 2 + 4\) becomes \(3x \leq 6\). Secondly, divide both sides by 3 to get the solution: \(x \leq \frac{6}{3}\) simplifies to \(x \leq 2\).
02

Solve the second inequality

The second inequality is \(4 x+5 \geq 5\). Following the same method, start by subtracting 5 from both sides of the inequality: \(4x \geq 5 - 5\) simplifies to \(4x \geq 0\). Then divide both sides by 4: \(x \geq \frac{0}{4}\) further simplifies to \(x \geq 0\).
03

Combine the results

We now have two inequalities which we can combine to get the solution. The first inequality says that \(x \leq 2\) and the second one states that \(x \geq 0\). Hence, the solution for x is within the interval \(0 \leq x \leq 2\).

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Key Concepts

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

Algebraic Inequalities
Algebraic inequalities are vital components of mathematics where we deal with the relative sizes of numbers or expressions.They tell us that one side of an inequality is greater than, less than, or possibly equal to another side, rather than giving a definitive answer as in algebraic equations.For example, in the inequality \(3x - 4 \leq 2\), the goal is to find all values of \(x\) that make the inequality true.

To solve an algebraic inequality, follow these steps:
  • Isolate the variable on one side of the inequality.
  • Perform inverse operations to simplify.
  • Whenever you multiply or divide by a negative number, the inequality sign flips its direction.
By solving inequalities, we gain insights into possible solutions rather than a single result, providing a range of potential values for the variable.
Compound Inequalities
Compound inequalities consist of two or more inequalities that are combined into one statement, often using the words "and" or "or."In the given exercise, we are dealing with an "and" situation with \(3 x-4 \leq 2\) and \(4 x+5 \geq 5\).This means both conditions must be satisfied at the same time.

To find the solution for compound inequalities:
  • Solve each inequality separately using standard techniques for algebraic inequalities.
  • Look for the overlap between solutions from both inequalities when using "and." For "or," you consider the union of all solutions.
In the provided example, the solution is the intersection where both conditions hold true, leading us to \(0 \leq x \leq 2\).This marks the values of \(x\) that satisfy both inequalities.
Interval Notation
Interval notation is a concise way of representing the set of solutions for inequalities.It uses brackets to show the endpoints of intervals, clearly indicating which values are included or excluded.

Here's how to understand interval notation:
  • Use square brackets \([ ]\) when an endpoint is included (also known as closed interval).
  • Use parentheses \(( )\) when an endpoint is excluded (open interval).
In the exercise solution, the interval \([0, 2]\) is used, reflecting that both endpoints are included in the solution set.This means the solution covers all numbers from 0 to 2, inclusive.Understanding interval notation helps in effectively communicating the solution sets for inequalities.

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