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Chapter 1: Problem 52

Solve the inequality. Express your answer in interval notation. $$3 x+1 \leq 7$$

Short Answer

Expert verified
The solution in the interval notation is \((-\infty, 2]\).

Step by step solution

01

Subtraction

Subtract 1 from both sides of the inequality to isolate the term containing x. This leads to \(3x \leq 6\).
02

Division

Divide both sides of the inequality by 3 to solve for x. This provides the result \(x \leq 2\).
03

Interval notation

Express the solution in interval notation. Since x can be any value that's 2 or less, it can be written as \((-\infty, 2]\).

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Key Concepts

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

Inequality Solving
Solving inequalities is similar to solving equations, but with a few differences. When we solve an inequality, we are looking to find all possible values of the variable that make the inequality true. In the example provided, the inequality to solve is \(3x + 1 \leq 7\). To solve it, we need to isolate the variable on one side of the inequality sign.
  • Start by subtracting 1 from both sides to get \(3x \leq 6\).
  • Then, divide each side by 3 to solve for \(x\), resulting in \(x \leq 2\).
One key point to remember is that if we multiply or divide both sides by a negative number, the direction of the inequality sign changes. However, in our example, we divided by a positive number, so the inequality sign remains \(\leq\). Understanding these rules will help ensure our solutions are correct.
Interval Notation
Interval notation is a way to represent a range of values, which are the solutions to an inequality. Once an inequality is solved, we express its solution in this form. From our example, we found \(x \leq 2\), meaning \(x\) can be 2 or any number less than 2. In interval notation, this is written as \((-\infty, 2]\).
  • The round bracket \((\) at \(-\infty\) indicates that \(-\infty\) is not a number but a concept, meaning values that extend indefinitely to the left.
  • The square bracket \(]\) at 2 signifies that 2 is included in the range of possible values for \(x\).
It's crucial to get familiar with these qualifiers, as they help in accurately depicting the solution set of an inequality.
Mathematical Operations
Mathematical operations are essential tools in solving inequalities. They include addition, subtraction, multiplication, and division鈥攁nd each serves a specific role. In our example, we applied subtraction and division:
  • Subtraction: Used to eliminate constant numbers from one side of the inequality. Here, we subtracted 1 to temporarily free \(x\) from its association with the constant.
  • Division: Employed to release \(x\) from its multiplicative bond with 3, resulting in an easier form \(x \leq 2\).
Understanding these operations helps manipulate expressions correctly. Also, a special note on division and multiplication: if we ever apply these operations with a negative number, it causes the inequality direction to flip, changing "greater than" to "less than," or vice versa. Always keep track of when to change inequality signs to avoid incorrect solutions.

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Most popular questions from this chapter

At a dew point of \(70^{\circ} \mathrm{F}\), the relative humidity, in percentage points, can be approximated by the linear function $$ R H(x)=-2.58 x+280 $$ where \(x\) represents the actual temperature. We assume that \(x \geq 70\), the dew point temperature. What is the range of temperatures for which the relative humidity is greater than or equal to \(50 \% ?\)

Solve the inequality. Express your answer in interval notation. $$\frac{x}{3} \leq \frac{2 x}{3}-1$$

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