S1 Mathematics Practice Book · Unit 3: Linear functions, equations and inequalities

Unit 3 Linear functions, equations and inequalities

Key unit competence: Apply linear equations and inequalities in solving real-life problems.

Key ideas

On a number line, numbers increase to the right. In the Cartesian plane a point is written $(x, y)$ : $x$ is measured along the horizontal axis, $y$ along the vertical axis, from the origin $O (0, 0)$ .
A linear function has the form $y = m x + c$ : $m$ is the gradient (slope) and $c$ is the y-intercept. Its graph is a straight line; two points are enough to draw it.
The x-intercept is found by putting $y = 0$ ; the y-intercept by putting $x = 0$ ; for $y = m x + c$ it is $(0, c)$ .
The gradient between $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ is $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ . Positive $m$ : the line rises; negative $m$ : it falls; $m = 0$ : horizontal.
Like terms have the same variables with the same powers; only like terms can be added or subtracted. When substituting, use brackets.
To solve a linear equation: expand brackets, clear fractions using the LCM, collect $x$ -terms on one side, then divide by the coefficient. Check by substituting.
Linear inequalities are solved like equations, but multiplying or dividing by a negative number reverses the sign. Solutions are shown on a number line (open circle for $<, >$ ; closed circle for $\leq, \geq$ ) or as intervals such as $(a, b)$ or $[a, b]$ .
Key words: "more than" $>$ , "less than" $<$ , "at least" $\geq$ , "at most" $\leq$ .

Formulas to know

y = m x + c

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

- x \leq 4 ⟺ x \geq - 4

Common misconceptions

"To solve $- 2 x < 6$ , divide by $- 2$ and get $x < - 3$ ." No: dividing by a negative reverses the sign: $x > - 3$ .
" $3 x + 2 x = 5 x^{2}$ ." No: $3 x + 2 x = 5 x$ ; we add coefficients of like terms, the power does not change.
"In $y = 2 x - 4$ , the x-intercept is $- 4$ ." No: $- 4$ is the y-intercept. Set $y = 0$ : $x = 2$ , so the x-intercept is $(2, 0)$ .

Worked examples

Example 1. Solve $y + \frac{2}{5} = \frac{3}{4} + \frac{y}{2}$ .

Solution

Multiply every term by 20: $20 y + 8 = 15 + 10 y$ .

$10 y = 7$ , so $y = \frac{7}{10}$ .

Example 2. Solve $3 x - 2 < 10 + x < 2 + 5 x$ and write the solution as an interval.

Solution

$3 x - 2 < 10 + x$ gives $2 x < 12$ , $x < 6$ . $10 + x < 2 + 5 x$ gives $8 < 4 x$ , $x > 2$ .

So $2 < x < 6$ , the interval $(2, 6)$ .

Practice questions

Easier and harder questions are mixed, just like in a real exam.

Find the x- and y-intercepts of $y = - 3 x + 6$ and sketch the line.
Show answer
y-intercept $(0, 6)$ . Put $y = 0$ : $3 x = 6$ , $x = 2$ , x-intercept $(2, 0)$ . Join the two points.
Common mistake: Giving the gradient $- 3$ as an intercept.
Solve $\frac{2 x - 1}{3} \leq \frac{x + 4}{2}$ and give the solution as an interval.
Show answer
Multiply by 6: $4 x - 2 \leq 3 x + 12$ , so $x \leq 14$ : interval $(- \infty, 14]$ .
Common mistake: Multiplying only the left side by 6.
Find the gradient of the line through $A (- 2, 5)$ and $B (4, - 1)$ .
Show answer
$m = \frac{- 1 - 5}{4 - ( - 2 )} = \frac{- 6}{6} = - 1$ .
Common mistake: Subtracting the coordinates in different orders on top and bottom.
The perimeter of a rectangle is 46 cm. The length is 5 cm more than the width. Find the dimensions.
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$2 (w + w + 5) = 46$ , $4 w + 10 = 46$ , $w = 9$ . Width 9 cm, length 14 cm.
Common mistake: Using $w + (w + 5) = 46$ and forgetting the factor 2.
If $p = - 5$ , $m = - 3$ and $n = 10$ , evaluate $\frac{2 mn}{p}$ .
Show answer
$\frac{2 \times ( - 3 ) \times 10}{- 5} = \frac{- 60}{- 5} = 12$ .
Common mistake: Losing a negative sign during substitution.
Peter buys tools at 2 500 FRW each and pays 3 000 FRW for transport. He can spend at most 28 000 FRW. How many tools can he buy?
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$2500 t + 3000 \leq 28000$ , $2500 t \leq 25000$ , $t \leq 10$ . At most 10 tools.
Common mistake: Writing $<$ for "at most".
Solve $4 (2 x - 1) - 3 (x + 2) = 5$ .
Show answer
$8 x - 4 - 3 x - 6 = 5$ , $5 x = 15$ , $x = 3$ .
Common mistake: Writing $- 3 (x + 2) = - 3 x + 6$ instead of $- 3 x - 6$ .
Solve and represent on a number line: $3 (x + 2) > 15$ .
Show answer
$3 x + 6 > 15$ , $3 x > 9$ , $x > 3$ . Open circle at 3 and arrow to the right.
Common mistake: Using a closed circle for a strict inequality.

Matching and drag-and-drop

Matching

Match each item in column A with the correct answer in column B. Write the letter next to each number.

Column A	Column B
1. A taxi charges 1 000 FRW plus 400 FRW per km. A ride costs 5 800 FRW. How long was it?	a. $(2, 0)$
2. If $x = 2$ , $y = 1$ and $z = 7$ , find $y + z - x$ .	b. $x = 4$
3. Solve $2 x + 3 = 11$ .	c. 12 km
4. Find the x-intercept of $y = 2 x - 4$ .	d. 6

Show answer

1 → c (12 km) · 2 → d (6) · 3 → b ( $x = 4$ ) · 4 → a ( $(2, 0)$ )

Step 1. $1000 + 400 d = 5800$ .
Step 2. $400 d = 4800$ , so $d = 12$ km.
$1 + 7 - 2 = 6$ .
Step 1. $2 x = 8$ .
Step 2. So $x = 4$ .
Step 1. Put $y = 0$ : $0 = 2 x - 4$ .
Step 2. So $x = 2$ .

Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
falls from left to right · $x = 21$ · 2, 3, 4 · $x = 27$
1. A line with negative gradient .
2. Which whole numbers satisfy $- 1 < 2 x - 3 \leq 5$ ? →
3. Solve $\frac{x}{3} - \frac{x - 1}{4} = 2$ . →
Show answer
1: falls from left to right · 2: 2, 3, 4 · 3: $x = 21$
- As $x$ increases, $y$ decreases.
- Add 3: $2 < 2 x \leq 8$ ; divide by 2: $1 < x \leq 4$ . Whole numbers: 2, 3, 4.
- Step 1. Multiply by 12: $4 x - 3 (x - 1) = 24$ .
  Step 2. So $4 x - 3 x + 3 = 24$ and $x = 21$ .
Matching
Match each item in column A with the correct answer in column B. Write the letter next to each number.
Column A Column B
1. Solve $\frac{x}{3} + \frac{x - 1}{2} = 4$ . a. $x \geq - 4$
2. Find the gradient of the line through $(1, 2)$ and $(3, 8)$ . b. $3 mn + 9 pq$
3. Simplify $8 mn + 6 pq - 5 mn + 3 pq$ . c. 3
4. Solve $5 - x \leq 9$ . d. $x = 5.4$
Show answer
1 → d ( $x = 5.4$ ) · 2 → c (3) · 3 → b ( $3 mn + 9 pq$ ) · 4 → a ( $x \geq - 4$ )
- Step 1. Multiply by 6: $2 x + 3 (x - 1) = 24$ .
  Step 2. $5 x - 3 = 24$ , so $5 x = 27$ and $x = 5.4$ .
- $m = \frac{8 - 2}{3 - 1} = \frac{6}{2} = 3$ .
- Collect like terms: $(8 - 5) mn + (6 + 3) pq$ .
- $- x \leq 4$ ; multiply by $- 1$ and reverse the sign: $x \geq - 4$ .
Drag and drop
Choose the correct word or phrase from the box for each gap. One word in the box is not needed.
$x = - 5$ · 69 · 3 · $x = - 13$
1. The sum of two numbers is 120 and their difference is 18. What is the larger number? →
2. Solve $2 (x - 3) - 3 (x + 1) = 4$ . →
3. The line $y = m x + 1$ passes through $(2, 7)$ . Find $m$ . →
Show answer
1: 69 · 2: $x = - 13$ · 3: 3
- Step 1. $x + (x + 18) = 120$ gives $x = 51$ .
  Step 2. So the larger is $51 + 18 = 69$ .
- Step 1. Expand: $2 x - 6 - 3 x - 3 = 4$ .
  Step 2. $- x - 9 = 4$ , so $- x = 13$ and $x = - 13$ .
- Step 1. $7 = 2 m + 1$ .
  Step 2. So $m = 3$ .

Column A	Column B
1. Solve $\frac{x}{3} + \frac{x - 1}{2} = 4$ .	a. $x \geq - 4$
2. Find the gradient of the line through $(1, 2)$ and $(3, 8)$ .	b. $3 mn + 9 pq$
3. Simplify $8 mn + 6 pq - 5 mn + 3 pq$ .	c. 3
4. Solve $5 - x \leq 9$ .	d. $x = 5.4$

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