Problem 28 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. \(\frac{3 x-2}{-5}<6\)

Step-by-Step Solution

Answer

The solution set is \( x > -\frac{28}{3} \) in interval notation \( \bigg( -\frac{28}{3}, \, \, \infty \bigg) \).

Step 1 - Isolate the variable

First, multiply both sides of the inequality by -5 to reverse the division and balance the inequality. Remember to flip the inequality sign when multiplying or dividing by a negative number. So, \[ \frac{3 x - 2}{-5} < 6 \] becomes \[ 3 x - 2 > -30 \].

Step 2 - Solve for x

Add 2 to both sides to isolate the term with the variable: \[ 3 x - 2 + 2 > -30 + 2 \] which simplifies to \[ 3 x > -28 \]. Next, divide both sides by 3: \[ x > -\frac{28}{3} \]

Step 3 - Write the solution in interval notation

Since we have \( x > -\frac{28}{3} \), the interval notation is \( \bigg( -\frac{28}{3}, \, \, \infty \bigg) \).

Step 4 - Graph the solution set

To graph the solution set, draw a number line. Place an open circle at \( -\frac{28}{3} \) to indicate that it is not included in the solution set, and shade the line to the right of \( -\frac{28}{3} \) to represent all values greater than \( -\frac{28}{3} \).

Key Concepts

Inequality Solutions

Solving inequalities is similar to solving equations but with one crucial difference: the inequality sign. Inequalities show the relationship between two expressions that are not necessarily equal, using symbols like <, >, ≤, and ≥.
The process to solve an inequality generally involves:

• Isolating the variable on one side of the inequality.
• Performing operations such as addition, subtraction, multiplication, or division.
• Reversing the inequality sign when multiplying or dividing by a negative number.

In the given exercise, \(\frac{3x-2}{-5} < 6\), we first multiplied both sides by -5 and flipped the inequality sign, resulting in \[3x-2 > -30\]. This is the crucial step many students might overlook. Continuing, we isolated x by adding 2 and then dividing by 3 to get \[x > -\frac{28}{3}\].
Understanding and following these rules ensures that you correctly solve inequalities.

Interval Notation

Interval notation is a shorthand way to express the set of solutions to an inequality. It describes the range of values that satisfy the inequality.
In interval notation, we use brackets and parentheses to denote closed and open boundaries:

• Use ( or ) for open intervals where end values are not included.
• Use [ or ] for closed intervals where end values are included.

For the inequality solution \( x > -\frac{28}{3} \), the interval notation is \( \big( -\frac{28}{3}, \, \, \infty \big) \) since \( -\frac{28}{3} \) is not included in the set, indicated by the parenthesis (open boundary). The inequality does not have an upper bound, so we use \( \infty \) to denote all larger numbers.
Remember, infinity and negative infinity are always accompanied by parentheses as they are not specific values but represent unbounded limits.

Graphing Inequalities

Graphing inequalities visually represents the solutions on a number line, making it easier to understand. Here are the steps to graph the inequality \( x > -\frac{28}{3} \):

• Draw a horizontal number line.
• Locate the point corresponding to \( -\frac{28}{3} \).
• Place an open circle at \( -\frac{28}{3} \) to indicate that this point is not included.
• Shade the line to the right of \( -\frac{28}{3} \) to represent all values greater than \( -\frac{28}{3} \).

This open circle and shading technique helps you quickly identify which section of the number line satisfies the inequality.
Graphing is a powerful tool because it translates abstract solutions into a concrete visual aid, highlighting the range of possible values.