Question

Solve: \(9-2 x \leq 4 x+1\)

Short answer

x \geq \frac{4}{3}

Step-by-step solution

- Move all terms involving x to one side

Subtract \(4x\) from both sides to keep the terms involving \(x\) on one side of the inequality:\[9 - 2x - 4x \leq 4x + 1 - 4x\]Simplifying this, we get:\[9 - 6x \leq 1\]

- Move constant terms to the other side

Subtract 9 from both sides to keep the constants on one side of the inequality:\[9 - 9 - 6x \leq 1 - 9\]Simplifying this, we get:\[-6x \-8\]

- Isolate the variable x

Divide both sides by \(-6\) to solve for \(x\). Remember to reverse the inequality sign when dividing by a negative number:\[\frac{-6x}{-6} \geq \frac{-8}{-6}\]Simplifying this, we get:\[x \geq \frac{4}{3}\]

Key concepts

Linear Inequalities

Linear inequalities are similar to linear equations but use inequality symbols instead of an equals sign. You'll often see symbols like \( <, \leq, >, \geq \). These symbols show the relationship between two expressions.
For instance, in the exercise \(9-2x \leq 4x + 1\), we need to determine the possible values of \(x\) that make the inequality true. Linear inequalities can be graphed on a number line to show all the solutions.

Inequality Manipulation

Manipulating inequalities involves similar steps to solving equations, but you must pay special attention to the inequality sign.
Step-by-step manipulation includes:

• Moving variable terms to one side. In the example \(9-2x \leq 4x + 1\), we subtracted \4x\ from both sides to get \(9 - 6x \leq 1\).
• Moving constant terms to the other side. We then subtracted 9 from both sides, resulting in \(-6x \leq -8\).
• Isolating the variable. We divided both sides by -6, and here remember to reverse the inequality sign to obtain \(x \geq \frac{4}{3}\).

These steps ensure you correctly manipulate the inequality to find the solution.

Algebraic Solutions

Solving inequalities algebraically involves finding ranges or specific values for the unknown that satisfy the inequality.
In our example, the final result is \(x \geq \frac{4}{3}\). This means any value of \(x\) that is equal to or greater than \( \frac{4}{3} \) will satisfy the original inequality.

• Always follow these steps:
• Isolate the variable term.
• Simplify the expressions.
• Perform valid operations that maintain the inequality balance.

By following these steps, you ensure that all algebraic solutions are accurate and complete.