Solving inequalities
Solving inequalities 1
Solve 𝑥 2 − 2𝑥 − 3 < 0
and write the set of integers that satisfy this inequality
Step 1
Solve the equation:
Let 𝑥 2 − 2𝑥 − 3 = 0
Factorise and solve (x – 3 )(x + 1) = 0
x = 3 , x = -1
These values are the critical values or the roots of the equation.
Step 2
5
Sketch the graph
-3 -2 -1
0
1
2
3
4
5
Step 3
Use the graph to solve the inequality:
Looking at the section of the graph below the x – axis as y < 0 so -1 < x < 3
The set of integers that satisfy this inequality is: { 0, 1, 2}
Practice
Solve the following inequalities and write out the integer set that satisfies the inequality.
a.
b.
c.
d.
x2
x2
x2
x2
+ 3x – 10 < 0
+ 8x + 15 ≤ 0
– 2x - 3 < 0
–4 ≤0
Extension
2x2 - 9x + 4 < 0
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Solving inequalities
Solutions:
a. x2 + 3x – 10 < 0
-5 < x < 2
{-4,-3,-2,-1,0,1}
b. x2 + 8x + 15 ≤ 0
-5 ≤ x ≤ -3
{-5,-4,-3}
c. x2 – 2x - 3 < 0
-1 < x < 3
{0,1,2}
d. x2 – 4 ≤ 0
-2 ≤ x ≤ 2
{-2,-1,0,1,2}
½<x<4
{1,2,3}
Extension:

2x2 - 9x + 4 < 0
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28258
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Solving inequalities
Solving inequalities 2
Solve 𝑥 2 − 2𝑥 − 3 > 0
Step 1
Solve the equation:
Let
𝑥 2 − 2𝑥 − 3 = 0
Factorise and solve
(x – 3 )(x + 1) = 0
x = 3 , x = -1
These values are the critical values or the roots of the equation.
Step 2
5
Sketch the graph
-3 -2 -1
0
1
2
3
4
5
Step 3
Use the graph to solve the inequality:
Looking at the section of the graph above the x – axis as y > 0 so
x > 3,
x < -1
Practice
Solve the following inequalities and write out the integer set that satisfies the inequality.
a. x2 – 3x + 2 > 0
b. x2 – 10x + 24 ≥ 0
c. x2 - 5x - 14 > 0
d. x2 – 9 ≥ 0
Extension
6x2 + 5x - 4 > 0
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28258
Page 3 of 4
Solving inequalities
Solutions:
a. x2 – 3x + 2 > 0
x< 1, x >2
b. x2 – 10x + 24 ≥ 0
x≤4 x≥6
c. x2 - 5x - 14 > 0
x < -2, x > 7
2
d. x – 9 ≥ 0
x ≤ -3, x ≥ 3
Extension

6x2 + 5x - 4 > 0
© www.teachitmaths.co.uk 2017
x<−
4
3
x >½
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