Question

1. |2x - 5|+x < 7

Explanation:

Step1: Isolate the absolute - value term

Subtract \(x\) from both sides of the inequality \(|2x - 5|+x<7\) to get \(|2x - 5|<7 - x\).

Step2: Consider two cases

Case 1: When \(2x - 5\geq0\) (i.e., \(x\geq\frac{5}{2}\))

The inequality \(|2x - 5|<7 - x\) becomes \(2x - 5<7 - x\).
Add \(x\) to both sides: \(2x+x - 5<7 - x+x\), which simplifies to \(3x-5 < 7\).
Add 5 to both sides: \(3x-5 + 5<7 + 5\), so \(3x<12\).
Divide both sides by 3: \(x < 4\).
Combining with the condition \(x\geq\frac{5}{2}\), we have \(\frac{5}{2}\leq x<4\).

Case 2: When \(2x - 5<0\) (i.e., \(x<\frac{5}{2}\))

The inequality \(|2x - 5|<7 - x\) becomes \(-(2x - 5)<7 - x\).
Expand the left - hand side: \(-2x + 5<7 - x\).
Add \(2x\) to both sides: \(-2x+2x + 5<7 - x+2x\), which simplifies to \(5<7 + x\).
Subtract 7 from both sides: \(5 - 7<7 + x-7\), so \(-2Combining with the condition \(x<\frac{5}{2}\), we have \(-2

Step3: Combine the solutions of the two cases

Combining the solutions from Case 1 and Case 2, we get \(-2

Answer:

\(-2