Linear Inequalities: math test solutions, Grade 9 – download the PDF

1 Solving inequalities / 4 pts

1. \(3x\le 18\), so \(x\le 6\). (1 pt)
2. \(-4x>16\); dividing by \(-4\) reverses the sign: \(x<-4\). (1.5 pt)
3. \(2x-6\ge 5x+6\) gives \(-12\ge 3x\), so \(x\le -4\). (1.5 pt)

2 Compound inequalities / 3 pts

1. Subtract 3: \(-8\le 2x<6\). Divide by 2: \(-4\le x<3\). Graph: closed circle at \(-4\), open circle at 3, segment shaded. (1.5 pt)
2. \(x<-3\) or \(x\ge 5\). (1.5 pt)

3 Absolute value equations / 4 pts

1. \(x-6=4\) gives \(x=10\); \(x-6=-4\) gives \(x=2\). (1 pt)
2. \(2|x+1|=12\), so \(|x+1|=6\). Then \(x=5\) or \(x=-7\). (2 pt)
3. An absolute value cannot be \(-1\): no solution. (1 pt)

4 Absolute value inequalities / 3 pts

1. \(-5\le x+2\le 5\), so \(-7\le x\le 3\). (1.5 pt)
2. \(2x-1>9\) gives \(x>5\); \(2x-1<-9\) gives \(x<-4\). So \(x<-4\) or \(x>5\). (1.5 pt)

5 Modeling / 4 pts

(a) \(4w-30\ge 250\) gives \(4w\ge 280\), so \(w\ge 70\): at least 70 wristbands. (2 pt)

(b) \(|p-12|\le 0.5\) gives \(11.5\le p\le 12.5\) pounds. In kilograms: \(11.5\times 0.4536\approx 5.22\) and \(12.5\times 0.4536\approx 5.67\), so about 5.22 kg to 5.67 kg. (2 pt)

6 True or false / 2 pts

1. False. Subtracting \(4x\) gives \(1<9\), always true, so every real number is a solution. (1 pt)
2. False. Dividing by \(-3\) reverses the sign: \(x\le -4\). (1 pt)