Question

Challenge Problem Solve: \(x-4<2 x-3 \leq \frac{x+5}{3}\)

Short answer

-1 < x \leq \frac{14}{5}

Step-by-step solution

Split the compound inequality

The compound inequality can be split into two separate inequalities: 1. Solve for the left inequality: \(x - 4 < 2x - 3\)2. Solve for the right inequality: \(2x - 3 \leq \frac{x + 5}{3}\)

Solve the left inequality

To solve \(x - 4 < 2x - 3\), follow these steps:1. Subtract \(x\) from both sides: \(-4 < x - 3\)2. Add 3 to both sides: \(-1 < x\) or \(x > -1\).

Solve the right inequality

To solve \(2x - 3 \leq \frac{x + 5}{3}\), follow these steps:1. Multiply both sides by 3 to eliminate the fraction: \(3(2x - 3) \leq x + 5\)2. Distribute the 3: \(6x - 9 \leq x + 5\)3. Subtract \(x\) from both sides: \(5x - 9 \leq 5\)4. Add 9 to both sides: \(5x \leq 14\)5. Divide by 5: \(x \leq \frac{14}{5}\).

Combine the solutions

From Step 2, we have \(x > -1\). From Step 3, we have \(x \leq \frac{14}{5}\). Combining these, the final solution is: \(-1 < x \leq \frac{14}{5}\).

Key concepts

Inequality Operations

When working with inequalities, the goal is similar to solving equations: isolate the variable on one side.However, there are a few additional rules to keep in mind.
For instance, if you multiply or divide both sides of an inequality by a negative number, the inequality sign must flip direction. This does not happen when working with equalities. The basic operations, like adding, subtracting, multiplying, or dividing both sides of the inequality by the same non-zero number, stay consistent.
Carefully managing these operations ensures that the inequality remains valid.

Solving Linear Inequalities

Solving linear inequalities involves a sequence of steps aimed at isolating the variable. Start by simplifying the inequality through basic algebraic operations:

• Perform addition or subtraction first to move constants to the other side.
• Use multiplication or division to isolate the variable.

For example, with the left inequality from the exercise, we had:
\(x - 4 < 2x - 3\).
1. Subtracting \(x\) from both sides results in \(-4 < x - 3\).
2. Then, by adding 3 to both sides we get \(-1 < x\), or equivalently \(x > -1\).
This sequence ensures that we isolate \(x\) effectively, giving the valid solution. Ensuring accuracy in these steps is crucial.

Fraction Elimination

Fractions in inequalities can be tricky. The key strategy is to eliminate the fraction early. Multiply all terms by the denominator of the fraction.
For the right inequality from the exercise:
\(2x - 3 \frac{x + 5}{3}\)
we start by multiplying every term by 3.
This gives us:
\(3(2x - 3) \ \ x + 5\),
which simplifies to:
\(6x - 9 \ \ x + 5\).
With the fraction out of the way, we can proceed with solving as we did with simpler linear inequalities.
This big first step simplifies our calculations and keeps the inequality easier to manage.

Inequality Combination

After solving separate inequalities, the next step is to combine them. Combining retains the valid ranges for the variable from each part.
From the first inequality, we had:
\(x > -1\).
From the second inequality, we had:
\(x \ 14 \ 5\).
Combining these gives us the range that satisfies both conditions:
\(-1 < x \ 14 \ 5\).
This represents the intersection of the two solution sets and provides a clear, consolidated range for \(x\).
Ensuring clarity in this final combination step guarantees the solution is accurate and all-encompassing.