Solve the inequality \(-9 \le 4x + 3 \lt 15\).
Question 1 Report
Solve the inequality \(-9 \le 4x + 3 \lt 15\).
Answer Details
A double inequality such as \(-9\le 4x+3\lt 15\) is really two statements about the same expression at once. The efficient method is to operate on all three parts simultaneously, so that \(4x+3\) is reduced to \(x\) while the outer numbers change with it.
Subtract 3 from each of the three parts:
\[-9-3\le 4x\lt 15-3\]\[-12\le 4x\lt 12\][M1]
Divide all three parts by the positive number 4, so both inequality signs keep their direction:
\[-3\le x\lt 3\]That is \(-3\le x\) [A1] and \(x\lt 3\) [A1].
The two ends behave differently and this must be preserved: \(x=-3\) is included because the original sign is \(\le\), giving \(4(-3)+3=-9\), while \(x=3\) is excluded because \(4(3)+3=15\) is not less than 15. Losing the distinction between \(\le\) and \(\lt\) is the usual way marks go here.
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