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Chapter 5: Problem 17

Solve each inequality and give a reason for each step in the solution. $$3(2 x+8) < 4(x-3)$$

Short Answer

Expert verified
x < -18

Step by step solution

01

- Distribute the constants

Distribute the constant 3 on the left side and the constant 4 on the right side of the inequality. 3(2x + 8) < 4(x - 3) will become: 6x + 24 < 4x - 12
02

- Move variable terms to one side

Subtract 4x from both sides to move the variable terms to one side.6x + 24 - 4x < 4x - 12 - 4x This simplifies to: 2x + 24 < -12
03

- Move constant terms to the other side

Subtract 24 from both sides to isolate the variable term.2x + 24 - 24 < -12 - 24 Simplifies to: 2x < -36
04

- Solve for the variable

Divide both sides by 2 to solve for x.\[ \frac{2x}{2} < \frac{-36}{2} \ x < -18 \]

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Key Concepts

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

distribution in algebra
Distribution in algebra involves distributing a constant across terms inside parentheses. This step can make the inequality easier to manage. For instance, we start with the expression:

\(3(2x + 8) < 4(x - 3)\)

  • Distribute the constant 3 to both terms inside the parentheses on the left side. This gives us 6x + 24.
  • Similarly, distribute the constant 4 to both terms inside the parentheses on the right side. This gives us 4x - 12.


After distribution, we get a simpler inequality:

\(6x + 24 < 4x - 12\)

Now, the inequality is in a form where we can isolate the variables.
isolating variables
Isolating variables helps us to solve equations and inequalities by getting the variable of interest by itself on one side of the equation or inequality. In our example, the inequality after distribution was:

\(6x + 24 < 4x - 12\)

  • First, we move the variable terms to one side by subtracting 4x from both sides. This simplifies to 2x + 24 < -12.
  • Next, move the constant terms to the other side. To do this, subtract 24 from both sides. Now, we get: 2x < -36.


We can now see only the variable term on one side of the inequality.
inequality manipulation
Inequality manipulation involves performing the same operation on both sides of the inequality to find the solution. After isolating variables, our simplified inequality is:

\(2x < -36\)

  • Divide both sides by 2 to solve for x, ensuring the direction of the inequality remains the same. This gives us:
\[ \frac{2x}{2} < \frac{-36}{2} \
x < -18 \]

  • Remember, if we had multiplied or divided by a negative number, we would reverse the inequality sign.
This step-by-step approach ensures we maintain the logical flow of inequality transformations and end up with the correct solution: x < -18.

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