Problem 14 - Algebra and Trigonometry

Question

Solve the linear inequality. Express the solution using interval notation and graph the solution set. $$ 3 x+11<5 $$

Step-by-Step Solution

Answer

x < -2, in interval notation: \((-\infty, -2)\).

Step 1: Isolate the Variable Term

First, we need to get the term containing the variable by itself on one side of the inequality. To do this, subtract 11 from both sides of the inequality: \[3x + 11 - 11 < 5 - 11\]This simplifies to:\[3x < -6\]

Step 2: Solve for x

Next, we want to solve for \(x\). Since \(3x\) means 3 times \(x\), divide both sides by 3 to solve for \(x\):\[\frac{3x}{3} < \frac{-6}{3}\]This simplifies to:\[x < -2\]

Step 3: Write the Solution in Interval Notation

Now that we have \(x < -2\), we express the solution in interval notation. The solution to \(x < -2\) is all real numbers less than -2, which is written as:\[(-\infty, -2)\]

Step 4: Graph the Solution Set

To graph the solution set on a number line, draw a number line with a point at -2. Since \(x\) is less than -2 and not equal to -2, draw an open circle at -2. Then shade the line to the left of -2 to indicate all values less than -2.

Key Concepts

Solving Inequalities

Solving inequalities is similar to solving equations, but with a key difference: instead of finding a single solution, we find a range of solutions that satisfy the inequality.
To solve an inequality such as \(3x + 11 < 5\), start by isolating the variable term just like you would in an equation.

• Subtract or add terms on both sides to move constants to the other side of the inequality.
• Next, divide or multiply both sides by a number to solve for the variable, making sure to reverse the inequality symbol if you multiply or divide by a negative number.

After following these steps, you should end with a statement like \(x < -2\). This indicates that any number less than -2 will make the inequality true.

Interval Notation

Once an inequality is solved, we use interval notation to express the solution succinctly. Interval notation is a shorthand way to describe a range of numbers.
In interval notation, we use parentheses \(()\) and brackets \([]\):

• Parentheses \(()\) indicate that a number is not included in the interval, generally used for inequalities with a "less than" or "greater than" symbol like \(x < -2\).
• Brackets \([]\) indicate an inclusive boundary, used with "less than or equal to" or "greater than or equal to" inequalities.

For the solution \(x < -2\), our interval notation would be \((-\infty, -2)\). This is because the solution includes all numbers less than -2, extending indefinitely towards negative infinity.

Graphing Inequalities

Graphing inequalities visually represents the range of solutions on a number line. This helps in understanding which sections of the number line satisfy the inequality.
Here's how to do it:

• First, draw a simple number line and identify the critical point on it using an open or closed circle. Use an open circle for strict inequalities (like \(<\) or \(>\)). For the inequality \(x < -2\), you place an open circle at -2.
• Next, shade the region of the number line that represents all solutions of the inequality. For \(x < -2\), shade to the left of -2, indicating all lower values are solutions.

This graphical representation allows for a quick visual check of possible solutions for the inequality.