Problem 40 - Algebra for College Students

Question

Solve each inequality. Graph the solution set, and write it using interval notation. $$ |-2 x-4| \geq 5 $$

Step-by-Step Solution

Answer

x < \frac{71}{150} Interval notation: (-∞, \frac{71}{150})

Step 1 - Clear the fractions

Multiply both sides of the inequality by 15 to eliminate the fractions. This is because 15 is the least common multiple (LCM) of 5 and 3: \[15 \times \frac{7}{5}(10x - 1) < 15 \times \frac{2}{3}(6x + 5)\]

Step 2 - Simplify

Simplify both sides after multiplying: \[3 \times 7 (10x - 1) < 5 \times 2 (6x + 5)\] This simplifies to: \[21 (10x - 1) < 10 (6x + 5)\]

Step 3 - Distribute the constants

Distribute the constants across the terms inside the parentheses: \[210x - 21 < 60x + 50\]

Step 4 - Collect like terms

Move the terms involving x to one side and the constant terms to the other side: \[210x - 60x < 50 + 21\] This simplifies to: \[150x < 71\]

Step 5 - Solve for x

Divide both sides of the inequality by 150 to isolate x: \[x < \frac{71}{150}\]

Step 6 - Graph the solution set

On a number line, draw an open circle at \(\frac{71}{150}\) and shade all the points to the left of this point, indicating all values less than \(\frac{71}{150}\).

Step 7 - Write in interval notation

The solution in interval notation is: \( (- ∞, \frac{71}{150}) \)

Key Concepts

inequality solutions

Solving inequalities is a critical skill in algebra. It often involves finding the set of values for a variable that makes an inequality true. In this exercise, we had to solve the inequality \(\frac{7}{5}(10 x-1)<\frac{2}{3}(6 x+5)\) following a structured step-by-step approach.
First, we cleared the fractions by multiplying the entire inequality by 15, the least common multiple (LCM) of 5 and 3. Multiplying both sides by 15 helped us remove the fractions, making it easier to work with the inequality.
Next, we simplified each side after multiplication, followed by distributing the constants across the terms inside the parentheses. This helped us collect like terms and rearrange the inequality to isolate the variable x. Finally, we divided both sides by 150 to solve for x, obtaining the solution as x < \(\frac{71}{150}\).
This step-by-step process is essential for tackling any inequality.

interval notation

Interval notation is a way of writing the set of all solutions to an inequality. It provides a concise and easily understandable method of displaying the range of values that satisfy an inequality. For instance, the solution to our inequality (x < \(\frac{71}{150}\) ) is written in interval notation as \((- \infty, \frac{71}{150})\).
Let's break it down:

• The round bracket \((\) indicates that the endpoint is not included, known as an 'open interval'.
• Negative infinity \((- \infty)\) means the interval extends indefinitely to the left.
• \(\frac{71}{150}\) is the specific value that x cannot reach, but gets close to.

Understanding interval notation helps simplify the presentation of solution sets for inequalities, making it easier to read and interpret.

graphing inequalities

Graphing inequalities involves representing the solution set of an inequality on a number line. It's a visual way to show all possible solutions. In our solution (x < \(\frac{71}{150}\) ), we graph it by:

• Drawing a number line.
• Placing an open circle at \(\frac{71}{150}\) to indicate that this value is not included in the solution set.
• Shading all points to the left of \(\frac{71}{150}\), showing every number less than \(\frac{71}{150}\).

This visual representation makes it clear which values for x satisfy the inequality. It helps us quickly understand the range and limit of the valid solutions.
Mastering graphing inequalities can give you a more intuitive understanding of the relationships within equations, enhancing your overall problem-solving skills.