Problem 18 - Algebra Form and Function

Question

Each of the inequalities can be solved by performing two operations on both sides. State the operations in order of use and solve the inequality. $$ \frac{4}{3} x \geq 2 x-3 $$

Step-by-Step Solution

Answer

Question: Solve the inequality and find the solution: \(\frac{4}{3}x \geq 2x - 3\). Answer: \(x \leq \frac{9}{2}\)

Step 1: Eliminate the fraction

To get rid of the fraction, multiply both sides of the inequality by 3: $$3 \cdot \frac{4}{3} x \geq 3(2 x-3)$$ This simplifies to: $$4x \geq 6x - 9$$

Step 2: Isolate x

Now, we combine the x terms and the constants. First, subtract \(6x\) from both sides to combine the x terms: $$4x - 6x \geq 6x - 6x - 9$$ which gives: $$-2x \geq -9$$ Next, to find the value of x, we will divide both sides by the coefficient of x, which is -2. Since we are dividing by a negative number, we need to change the inequality sign: $$\frac{-2x}{-2} \leq \frac{-9}{-2}$$ This simplifies to: $$x \leq \frac{9}{2}$$ So, the solution to the inequality is \(x \leq \frac{9}{2}\).

Key Concepts

Fraction Elimination in Inequalities

Dealing with fractions in inequalities can be intimidating, but not difficult with a simple operation: elimination. To make an inequality easier to solve, it's beneficial to remove fractions early on. This is achieved by multiplying both sides of the inequality by the denominator of the fraction you want to eliminate. For example, if you have an expression with the fraction \(\frac{4}{3}x \geq 2x - 3\), multiplying the entire inequality by 3 will clear the fraction:

• The left side \(3 \times \frac{4}{3} x = 4x\)
• The right side \(3 \times (2x - 3) = 6x - 9\)

Once the fractions are eliminated, you are left with an inequality without fractions \(4x \geq 6x - 9\), making it more straightforward to solve in subsequent steps.

Isolation of Variables

Once fractions are dealt with, the next step is to isolate the variable you are solving for—in this case, \(x\). Isolating the variable involves moving all terms involving \(x\) to one side of the inequality. Start by combining like terms. Using our example \(4x \geq 6x - 9\), you need to get all the \(x\)-terms on one side:

• Subtract \(6x\) from both sides to remove \(x\)-terms from the right side: \(4x - 6x \geq 6x - 6x - 9\).
• This simplifies the inequality to \(-2x \geq -9\).

Next, divide by the coefficient of \(x\), which in this example is \(-2\), to solve for \(x\). Remember, when you divide by a negative number, reverse the inequality sign:

• \(\frac{-2x}{-2} \leq \frac{-9}{-2}\)
• The solution is \(x \leq \frac{9}{2}\)

The variable is now isolated, and your inequality is solved!

Understanding Inequality Properties

Inequalities have unique properties that distinguish them from equations. A critical property to remember is that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. This step is necessary to maintain the true statement of the inequality. In our example, when dividing both sides by \(-2\) to isolate \(x\), the inequality sign \(\geq\) turned into \(\leq\).

• This rule ensures that the inequality's logic remains intact.
• Without this reversal, the solution will represent a different condition.

Additionally, inequalities are useful beyond traditional equations, as they show a range of possible solutions rather than a single outcome. Understanding these properties helps in both solving and interpreting inequalities correctly.