Question

Solve \(5(x-6)<2(x-9)\) A. \(x<16\) B. \(x>4\) C. \(x<3\) D. \(x<4\)

Short answer

D. \(x < 4\)

Step-by-step solution

Distribute the constants

Distribute the constants inside the parentheses on both sides of the inequality: On the left side: \[5(x - 6) = 5x - 30\]On the right side: \[2(x - 9) = 2x - 18\]So the inequality becomes: \[5x - 30 < 2x - 18\]

Move variable terms to one side

Subtract 2x from both sides of the inequality to get all variable terms on one side: \[5x - 30 - 2x < 2x - 18 - 2x\]Simplify to get: \[3x - 30 < -18\]

Isolate the variable

Add 30 to both sides of the inequality to isolate the term with the variable: \[3x - 30 + 30 < -18 + 30\]Simplify to get: \[3x < 12\]

Solve for x

Divide both sides of the inequality by 3 to solve for x: \[\frac{3x}{3} < \frac{12}{3}\]Simplify to get: \[x < 4\]

Choose the correct option

Compare the solution to the provided options. The solution \(x < 4\) matches option D.

Key concepts

Solving Inequalities

Solving inequalities is similar to solving equations, but with an added twist. Inequalities show a range of possible solutions rather than just one.
When solving inequalities, follow these general steps:

• Distribute any constants.
• Move variable terms to one side.
• Isolate the variable.
• Simplify the equation.

We'll walk through each step, so you can get a solid understanding of how to solve inequalities like a pro!

Distributing Constants

The first step in solving the given inequality, \(5(x-6)<2(x-9)\), is to distribute the constants. Distributing means multiplying the constants through the parentheses.

• On the left side: \(5(x - 6) = 5x - 30\)
• On the right side: \(2(x - 9) = 2x - 18\)

Now, our inequality looks like this: \(5x - 30 < 2x - 18\). Distributing ensures all terms inside the parentheses are considered and simplified first.

Isolating the Variable

Once the constants are distributed, the next goal is to isolate the variable (usually \(x\)). To do this:

• Subtract 2x from both sides to keep all variable terms on one side: \(5x - 30 - 2x < 2x - 18 - 2x\). Simplifies to: \(3x - 30 < -18\).
• To isolate \(3x\), add 30 to both sides: \(3x - 30 + 30 < -18 + 30\). Simplifies to: \(3x < 12\).

Always perform the same operation on both sides of the inequality to keep it balanced and accurate.

Simplifying Equations

The final step is simplifying the equation to find the value of the variable. In our simplified inequality, \(3x < 12\),

• To solve for \(x\), divide both sides by 3: \(\frac{3x}{3} < \frac{12}{3}\).
• This results in: \(x < 4\).

After simplifying, we see that \(x < 4\) is the solution, which matches option D.
Remember, simplifying equations involves basic arithmetic operations to isolate the variable and solve the inequality completely.