Problem 43 - Beginning and Intermediate Algebra

Question

Solve each inequality, and graph the solution set. $$ \frac{3}{2 x-1}<2 $$

Step-by-Step Solution

Answer

The solution set is \( x < \frac{1}{2} \) or \( x > \frac{5}{4} \).

Step 1: Isolate the Fraction

To solve the inequality \( \frac{3}{2x-1} < 2 \), start by isolating the fraction on one side. The fraction is already isolated, so we can proceed to the next step.

Step 2: Set Up Equivalent Inequality

Create an equivalent inequality by multiplying both sides of the inequality by \( 2x - 1 \). Note that since we are dealing with an inequality, we need to consider two cases: when \( 2x - 1 > 0 \) and when \( 2x - 1 < 0 \). This gives us two separate inequalities to solve.

Step 3: Case 1 - \( 2x - 1 > 0 \)

First, solve for \( 2x - 1 > 0 \). This simplifies to \( 2x > 1 \), so \( x > \frac{1}{2} \). Now solve the inequality \( \frac{3}{2x-1} < 2 \) assuming \( 2x - 1 > 0 \). Multiply both sides by \( 2x - 1 \): \( 3 < 2(2x - 1) \). Simplify to get \( 3 < 4x - 2 \). Adding 2 to both sides: \( 5 < 4x \). Finally, dividing both sides by 4 gives: \( \frac{5}{4} < x \). Combining with \( x > \frac{1}{2} \), we get \( x > \frac{5}{4} \).

Step 4: Case 2 - \( 2x - 1 < 0 \)

Now solve for \( 2x - 1 < 0 \). This simplifies to \( 2x < 1 \), so \( x < \frac{1}{2} \). Next, solve the inequality \( \frac{3}{2x-1} < 2 \) assuming \( 2x - 1 < 0 \). Because \( 2x - 1 \) is negative, multiplying the inequality by \( 2x - 1 \) reverses the inequality: \( 3 > 2(2x - 1) \). Simplify to get \( 3 > 4x - 2 \). Adding 2 to both sides: \( 5 > 4x \). Finally, dividing both sides by 4 gives: \( \frac{5}{4} > x \). Combining with \( x < \frac{1}{2} \), we get \( x < \frac{1}{2} \).

Step 5: Combine Solutions

From Case 1 and Case 2, the combined solution set is \( x < \frac{1}{2} \) or \( x > \frac{5}{4} \).

Step 6: Graph the Solution

Draw a number line and mark the points \( \frac{1}{2} \) and \( \frac{5}{4} \). Shade the region to the left of \( \frac{1}{2} \) and the region to the right of \( \frac{5}{4} \). Do not include \( \frac{1}{2} \) and \( \frac{5}{4} \) themselves, indicating these points are not part of the solution.

Key Concepts

inequality solutions

Inequality solutions involve finding the values of a variable that make the inequality true. When solving inequalities, always perform similar steps as you would when solving equations, but remember to flip the inequality sign when multiplying or dividing by a negative number.

Consider the given inequality \(\frac{3}{2x - 1} < 2 \).

• First, isolate the fraction, which means making sure it's already on one side. In our case, it is.
• Secondly, create an equivalent inequality by multiplying both sides by the denominator (considering both when the denominator is positive and negative).
• This gives us two cases to solve: one where the denominator is positive and one where it's negative.

These steps ensure you account for all possible values for x and prevent any missed solutions.

isolating fractions

Isolating fractions means getting the fraction alone on one side of the inequality or equation before you begin manipulating it. This makes solving easier.

Here’s how we isolated the fraction in our example:

\(\frac{3}{2x-1} < 2 \)

• The fraction \(\frac{3}{2x-1}\) is already isolated.
• The next important step is to create an equivalent inequality by multiplying both sides by \(({2x-1})\). Remember to consider both cases where \(({2x-1})\) is greater than 0 and less than 0.
• This results in two separate inequalities to solve.

Multiplying inequalities adds specific rules. When multiplying by a negative, reverse the inequality sign, ensuring the correct solution set.

graphing inequalities

Graphing inequalities involves visually representing the solution set on a number line or coordinate plane. This helps understand which values satisfy the inequality.

For the inequality \(\frac{3}{2x-1} < 2\):

• After solving the inequalities, we end up with two separate solutions: \( x < \frac{1}{2} \) and \( x > \frac{5}{4} \).
• On a number line, mark the points \( \frac{1}{2} \) and \( \frac{5}{4} \).
• Shade the regions to the left of \( \frac{1}{2} \) and to the right of \( \frac{5}{4} \), since these are the solution sets.
• These points themselves aren't included in the solutions, indicated using open circles on the number line.

Graphing helps in visualizing which parts of the number line satisfy the inequality, offering a clearer understanding of the solution set.