Problem 55

Question

Solve each compound inequality. $$-11<2 x-1 \leq-5$$

Step-by-Step Solution

Verified

Answer

The solution to the compound inequality is $-5 < x \leq -2$.

1Step 1: Split the Compound Inequality

Divide the compound inequality into two separate ones: i. $-11 < 2x - 1$ and ii. $2x - 1 \leq -5$

2Step 2: Solve the first inequality

For the first inequality $-11 < 2x - 1$, add 1 to both sides to isolate $2x$, leading to $-10 < 2x$. Then, divide both sides by 2 to solve for $x$, so $x > -5$.

3Step 3: Solve the second inequality

For the second inequality $2x - 1 \leq -5$, add 1 to both sides to isolate $2x$, leading to $2x \leq -4$. Then, divide both sides by 2 to solve for $x$, so $x \leq -2$.

4Step 4: Combine Solutions

Now, combine the solutions of the two inequalities to find the intersection. The previous two steps gave us $x > -5$ and $x \leq -2$, both hold true for the solution which is $-5 < x \leq -2$

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