Question

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match each compound inequality on the left to the graph that represents its solution on the right.
-14 ≤ 6x + 4 < 16
-4x + 3 > -9 and -6x ≤ 12
-4x + 3 < -9 or 6x ≤ -12
clear
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Explanation:

Step1: Solve the first compound inequality \(-14\leq6x + 4<16\)

Subtract \(4\) from all parts: \(-14-4\leq6x+4 - 4<16 - 4\), so \(-18\leq6x<12\).
Divide by \(6\): \(\frac{-18}{6}\leq\frac{6x}{6}<\frac{12}{6}\), which simplifies to \(-3\leq x<2\).

Step2: Solve the second compound inequality \(-4x + 3>-9\) and \(-6x\leq12\)

For \(-4x+3>-9\), subtract \(3\): \(-4x>-9 - 3\), so \(-4x>-12\). Divide by \(-4\) (reverse the inequality sign): \(x < 3\).
For \(-6x\leq12\), divide by \(-6\) (reverse the inequality sign): \(x\geq - 2\). The solution is \(-2\leq x<3\).

Step3: Solve the third compound inequality \(-4x + 3<-9\) or \(6x\leq - 12\)

For \(-4x+3<-9\), subtract \(3\): \(-4x<-9 - 3\), so \(-4x<-12\). Divide by \(-4\) (reverse the inequality sign): \(x>3\).
For \(6x\leq - 12\), divide by \(6\): \(x\leq - 2\). The solution is \(x\leq - 2\) or \(x>3\).

Answer:

\(-14\leq6x + 4<16\) matches the first graph (with closed - circle at \(-3\) and open - circle at \(2\)).
\(-4x + 3>-9\) and \(-6x\leq12\) matches the second graph (with closed - circle at \(-2\) and open - circle at \(3\)).
\(-4x + 3<-9\) or \(6x\leq - 12\) matches the third graph (with closed - circle at \(-2\) and open - circle at \(3\) where the line is on the two ends).