Linear Inequalities: practice solutions, Grade 9 – download the PDF

1 Words to symbols ★★★

Match each phrase to a symbol: “at least” and “minimum” mean \(\ge\); “fewer than” and “more than” are strict; “no more than” and “at most” mean \(\le\).

1. \(x\ge 18\)
2. \(x<12\)
3. \(x\le 40\)
4. \(x>7\)
5. \(x\le 9\)
6. \(x\ge 5\)

2 Testing solutions ★★★

Substitute each number.

• \(x=5\): \(3(5)-4=11\), and \(11<11\) is false.
• \(x=4\): \(8<11\), true.
• \(x=6\): \(14<11\), false.
• \(x=-2\): \(-10<11\), true.

The solutions are 4 and \(-2\). (In fact, solving gives \(x<5\).)

3 One-step inequalities ★★★

1. Subtract 7: \(x\le 5\).
2. Add 4: \(x>3\).
3. Divide by 6: \(x<9\).
4. Multiply by 5: \(x\ge 15\).

No sign changes, because we only used positive numbers.

4 Flip the sign ★★★

In every part we divide or multiply by a negative number, so the sign reverses.

1. \(x>-6\)
2. \(x\le -5\)
3. Multiply by \(-4\): \(x<-8\).
4. \(x\ge 3\)

Check (a) with \(x=0\): \(0<18\), true, and 0 is greater than \(-6\).

5 Reading a graph ★★★

The first graph has a closed circle at \(-2\) and shading to the right: \(x\ge -2\). The second has an open circle at 3 and shading to the left: \(x<3\).

6 Drawing graphs ★★★

(a) Open circle at 1, shade to the right. (b) Closed circle at \(-3\), shade to the left.

7 Two-step inequalities ★★★

1. \(2x>8\), so \(x>4\).
2. \(5x\le 25\), so \(x\le 5\).
3. \(\dfrac{x}{3}<4\), so \(x<12\).
4. \(4x\ge -12\), so \(x\ge -3\).

8 Variables on both sides ★★★

1. Subtract \(2x\): \(3x-7>8\). Add 7: \(3x>15\). So \(x>5\).
2. Distribute: \(3x+12\le 2x+9\). Subtract \(2x\): \(x+12\le 9\). So \(x\le -3\).
3. Subtract \(4x\): \(7-6x\ge 25\). Subtract 7: \(-6x\ge 18\). Divide by \(-6\) and flip: \(x\le -3\). Check \(x=-3\): both sides equal 13.

9 All numbers or none ★★★

1. Distribute: \(2x+6>2x+1\). Subtract \(2x\): \(6>1\), which is always true. All real numbers are solutions.
2. Subtract \(3x\): \(-5\le -8\), which is false. No solution.
3. Subtract \(x\): \(5\ge 5\), always true. All real numbers.

10 Spot the error ★★★

The student divided by a negative number without reversing the sign. Correct: \(-2x>8\) gives \(x<-4\). Check \(x=-5\): \(10+1=11>9\), true; check \(x=0\): \(1>9\), false.

11 Compound with and ★★★

1. Add 1 to every part: \(-6<3x\le 12\). Divide by 3: \(-2
2. Divide every part by \(-2\), reverse both signs, and reverse the order: \(-2\ge x>-5\), i.e. \(-5

12 Compound with or ★★★

1. \(x<-2\) or \(x\ge 7\).
2. \(2x>6\) gives \(x>3\); \(-x\ge 1\) gives \(x\le -1\). So \(x\le -1\) or \(x>3\).

13 Absolute value equations ★★★

1. \(x=8\) or \(x=-8\).
2. \(x+3=5\) gives \(x=2\); \(x+3=-5\) gives \(x=-8\).
3. \(3x-6=12\) gives \(x=6\); \(3x-6=-12\) gives \(x=-2\).
4. An absolute value cannot equal \(-4\), so there is no solution.

14 Absolute value inequalities ★★★

1. \(-6
2. \(x\le -4\) or \(x\ge 4\).
3. \(-3\le x-2\le 3\), so \(-1\le x\le 5\).
4. \(x+1>2\) gives \(x>1\); \(x+1<-2\) gives \(x<-3\). So \(x<-3\) or \(x>1\).

15 School supplies budget ★★★

Let \(n\) be the number of notebooks. Then \(22.50+3.75n\le 60\). Subtract 22.50: \(3.75n\le 37.50\). Divide by 3.75: \(n\le 10\).

Mia can buy at most 10 notebooks. Check: \(22.50+37.50=60\), exactly her budget.

16 Fractions and decimals ★★★

1. Multiply every term by 12: \(4(2x-1)-3(x+2)\ge 12\). Expand: \(8x-4-3x-6\ge 12\), so \(5x-10\ge 12\), \(5x\ge 22\), \(x\ge 4.4\).
2. Multiply by 10: \(4x+12

17 Harder absolute equations ★★★

1. Subtract 4 and divide by 3: \(|2x-1|=5\). Then \(2x-1=5\) gives \(x=3\); \(2x-1=-5\) gives \(x=-2\).
2. Two expressions have the same absolute value when they are equal or opposites. Case 1: \(3x-2=x+6\), so \(x=4\). Case 2: \(3x-2=-(x+6)\), so \(4x=-4\) and \(x=-1\). Check: \(|10|=|10|\) and \(|-5|=|5|\).
3. \(5-2x=7\) gives \(x=-1\); \(5-2x=-7\) gives \(x=6\).

18 Harder absolute inequalities ★★★

1. \(2|x-4|<8\), so \(|x-4|<4\). Then \(-4
2. Divide by \(-3\) and flip: \(|x+1|\ge 4\). Then \(x+1\ge 4\) gives \(x\ge 3\); \(x+1\le -4\) gives \(x\le -5\). So \(x\le -5\) or \(x\ge 3\).
3. Add 5: \(|4x-3|>5\). Then \(4x-3>5\) gives \(x>2\); \(4x-3<-5\) gives \(x<-\tfrac12\). So \(x<-0.5\) or \(x>2\).

19 Bottle tolerance ★★★

(a) \(|v-500|\le 6\).

(b) \(-6\le v-500\le 6\), so \(494\le v\le 506\).

(c) 507 is greater than 506, so the bottle is not acceptable.

(d) \(494\times 0.0338\approx 16.7\) and \(506\times 0.0338\approx 17.1\), so about 16.7 to 17.1 fluid ounces.

20 Reaching a test average ★★★

The sum of the first four is \(82+90+78+88=338\). We need \(\dfrac{338+s}{5}\ge 85\). Multiply by 5: \(338+s\ge 425\), so \(s\ge 87\). Since scores cannot exceed 100: \(87\le s\le 100\).

Jordan needs a score between 87 and 100.

21 Bakery profit ★★★

Let \(m\) be the number of muffins. Profit is revenue minus costs: \(2.75m-(1.10m+120)\ge 300\). Simplify: \(1.65m-120\ge 300\), so \(1.65m\ge 420\) and \(m\ge 254.5\ldots\)

Because \(m\) must be a whole number, the bakery must sell at least 255 muffins. Check: 254 muffins give \(1.65\times 254=419.10<420\); 255 give \(420.75\ge 420\).

22 Comparing two plans ★★★

Let \(m\) be the number of movies. Plan B is cheaper when \(4+1.6m<12\). Subtract 4: \(1.6m<8\). Divide by 1.6: \(m<5\).

(a) Plan B is cheaper for fewer than 5 movies, i.e. 0, 1, 2, 3, or 4 movies. (b) At \(m=5\): \(4+8=12\), so the prices are equal.