The Lumenary · Grades 6–8

Linear equations and inequalities

Name: ______________________

Date: ______________________

You will learn to

Solve linear equations with the variable on both sides, including fractions, brackets and negative coefficients, and check the solutions
Explain why adding the same number to both sides, or multiplying both sides by the same non-zero number, gives an equivalent equation
Recognise equations with exactly one solution, no solution or infinitely many solutions, and prove that no linear equation has exactly two
Solve linear inequalities, reversing the inequality when multiplying or dividing by a negative number, and prove why the reversal happens
Model situations with linear equations and inequalities and interpret the solutions

1.
Solve $5x - 7 = 2x + 8$ .
2.
Solve $\dfrac{x + 2}{3} = \dfrac{x - 1}{2}$ .
3.
Which of these equations have no solution? (Choose all that apply.)
- A. $3(x + 2) = 3x + 6$
- B. $3(x + 2) = 3x + 5$
- C. $2x + 1 = 2x - 1$
- D. $4x = 0$
- E. $x + 1 = 1 + x$
4.
Ravi starts with the equation $x = 5$ . He multiplies both sides by $0$ , gets $0 = 0$ , and concludes that every number is a solution. What went wrong?
- A.Multiplying by $0$ is not reversible, so $0 \cdot x = 0 \cdot 5$ is not equivalent to $x = 5$ .
- B.Nothing: $0 = 0$ is true, so every number is a solution of $x = 5$ .
- C.He should have multiplied only one side by $0$ .
- D. $0 \cdot 5$ is not $0$ .
5.
Solve $4 - 2x > 10$ .
- A. $x < -3$
- B. $x > -3$
- C. $x < 3$
- D. $x > 7$
6.
In Manila, one phone-repair shop charges $150$ pesos plus $80$ pesos per hour of work. Another charges $50$ pesos plus $100$ pesos per hour. For how many hours of work do the two shops charge the same?
7.
Ama has $50$ cedis. She buys a bus ticket for $8$ cedis and wants to spend the rest on mangoes at $4$ cedis each. What is the largest number of mangoes she can buy?
8.
Which Lean statement correctly says "multiplying both sides of an inequality by a negative number reverses it"?
- A.∀ p q c : Rat, c < 0 → p < q → c * q < c * p
- B.∀ p q c : Rat, p < q → c * q < c * p
- C.∀ p q c : Rat, c < 0 → p < q → c * p < c * q
- D.∃ p q c : Rat, c < 0 ∧ p < q ∧ c * q < c * p
9.
Write solve_linear(a, b, c, d) for the equation $ax + b = cx + d$ with integer coefficients. Return the solution as a Fraction if there is exactly one, the string "none" if there is no solution, and "all" if every number is a solution.
Computational — Practice
10.
Emma, a student in Melbourne, claims she has found a linear equation $ax + b = cx + d$ whose solutions are exactly $x = 1$ and $x = 2$ . Which argument shows this is impossible?
- A.Both values solve $(a - c)x = d - b$ ; subtracting gives $(a - c) \cdot 1 = 0$ , so $a = c$ , then $d = b$ , and every number is a solution.
- B.A linear equation always has exactly one solution.
- C.We can test $x = 3$ : it will not be a solution.
- D.Equations with two solutions must contain $x^2$ , so her equation is not linear.

Answer key — Linear equations and inequalities

1.
$5x - 7 = 2x + 8 \Rightarrow 3x - 7 = 8 \Rightarrow 3x = 15 \Rightarrow x = 5$ . Check: $5 \cdot 5 - 7 = 18$ and $2 \cdot 5 + 8 = 18$ .
2.
Multiplying by $6$ : $2(x + 2) = 3(x - 1)$ , so $2x + 4 = 3x - 3$ and $x = 7$ . Check: $\frac{9}{3} = 3$ and $\frac{6}{2} = 3$ .
3.
In standard form: (a) $0x = 0$ , every $x$ ; (b) $0x = -1$ , no solution; (c) $0x = -2$ , no solution; (d) $4x = 0$ , exactly one solution $x = 0$ ; (e) $0x = 0$ , every $x$ . So (b) and (c) have no solution.
4.
Multiplying both sides by $c$ gives an equivalent equation only when $c \neq 0$ , because the step must be undone by multiplying by $\frac{1}{c}$ . Multiplying by $0$ turns every equation into $0 = 0$ and destroys the information: $x = 5$ has one solution, $0 \cdot x = 0 \cdot 5$ has infinitely many.
5.
$4 - 2x > 10 \Rightarrow -2x > 6 \Rightarrow x < -3$ , reversing the inequality because we divided by $-2$ . Check: $x = -4$ gives $12 > 10$ (true) and $x = 0$ gives $4 > 10$ (false).
6.
$150 + 80h = 50 + 100h \Rightarrow 100 = 20h \Rightarrow h = 5$ . Both shops charge $550$ pesos for $5$ hours. For less work the second shop is cheaper; for more, the first.
7.
$8 + 4m \leq 50 \Rightarrow 4m \leq 42 \Rightarrow m \leq 10.5$ . Since $m$ is a whole number, the largest possible value is $m = 10$ (costing $48$ cedis); $11$ mangoes would need $52$ .
8.
∀ p q c : Rat, c < 0 → p < q → c * q < c * p. It quantifies over all numbers, assumes c < 0 and p < q, and concludes the reversed inequality. Dropping c < 0, or keeping the direction, gives false statements; using ∃ gives a much weaker one.
9.
```
from fractions import Fraction

def solve_linear(a, b, c, d):
    coeff, rhs = a - c, d - b
    if coeff != 0:
        return Fraction(rhs, coeff)
    return "all" if rhs == 0 else "none"
```
This follows the theorem on $ax = b$ exactly: one solution when the coefficient is non-zero; otherwise the equation reads $0 = d - b$ , which is either always true or never true.
10.
From $(a - c) \cdot 2 = d - b$ and $(a - c) \cdot 1 = d - b$ , subtracting gives $a - c = 0$ , and then $d - b = 0$ . The equation is then $0 \cdot x = 0$ , solved by every number — so it cannot have exactly the two solutions $1$ and $2$ .