Linear Equations and Inequalities: Question 1

Worked solution

Strategy

For a linear equation with brackets and the unknown on both sides, work in a fixed order:

1. Expand every bracket (multiply each term inside by the number outside).
2. Simplify each side by collecting like terms.
3. Collect the $x$-terms on one side and the number terms on the other.
4. Solve by dividing.
5. Check by substituting your answer back in.

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Part (a): $4(x + 3) = 2(x + 9)$

Step 1: Expand both brackets. Multiply both terms inside each bracket:

$4(x+3) = 4x + 12, \qquad 2(x+9) = 2x + 18$

So the equation becomes

$4x + 12 = 2x + 18.$

Step 2: Collect the $x$-terms. Subtract $2x$ from both sides (this keeps the $x$-coefficient positive):

$4x - 2x + 12 = 18 \;\;\Longrightarrow\;\; 2x + 12 = 18.$

Step 3: Collect the numbers. Subtract $12$ from both sides:

$2x = 18 - 12 = 6.$

Step 4: Solve. Divide both sides by $2$:

$x = \frac{6}{2} = 3.$

Step 5: Check. $\;$ LHS $= 4(3+3) = 4 \times 6 = 24$; $\;$ RHS $= 2(3+9) = 2 \times 12 = 24.$ $\checkmark$

$\boxed{x = 3}$

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Part (b): $20 - 3(x - 1) = 2(x + 4) + 5$

Step 1: Expand each bracket carefully. On the left the bracket is multiplied by $-3$, so watch the signs:

$-3(x - 1) = -3x + 3 \quad(\text{since } -3 \times -1 = +3).$

Left-hand side: $\;20 - 3x + 3 = 23 - 3x.$

Right-hand side: $\;2(x + 4) + 5 = 2x + 8 + 5 = 2x + 13.$

The equation is now

$23 - 3x = 2x + 13.$

Step 2: Collect the $x$-terms. The $x$-term on the left is negative, so add $3x$ to both sides:

$23 = 2x + 3x + 13 \;\;\Longrightarrow\;\; 23 = 5x + 13.$

Step 3: Collect the numbers. Subtract $13$ from both sides:

$23 - 13 = 5x \;\;\Longrightarrow\;\; 10 = 5x.$

Step 4: Solve. Divide both sides by $5$:

$x = \frac{10}{5} = 2.$

Step 5: Check. $\;$ LHS $= 20 - 3(2 - 1) = 20 - 3 = 17$; $\;$ RHS $= 2(2 + 4) + 5 = 12 + 5 = 17.$ $\checkmark$

$\boxed{x = 2}$

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Alternative for part (b)

You could instead gather the $x$-terms on the left. From $23 - 3x = 2x + 13$, subtract $2x$ and subtract $23$:

$-3x - 2x = 13 - 23 \;\;\Longrightarrow\;\; -5x = -10 \;\;\Longrightarrow\;\; x = \frac{-10}{-5} = 2.$

Same answer; moving the unknowns to whichever side keeps the coefficient positive simply avoids a final division by a negative number.