Download Linear Equations and Inequalities

Linear Equations
EXAMPLES:
1. Solve the equation 7x − 4 = 3x + 8.
Solution: We have
7x − 4 = 3x + 8
7x − 4 + 4 = 3x + 8 + 4
7x − 4 = 3x + 8
7x = 3x + 12
7x − 3x = 8 + 4
7x − 3x = 3x + 12 − 3x
or, in short,
4x = 12
4x = 12
x=
4x
12
=
4
4
x=3
12
=3
4
2. Solve the equation −5(x − 4) + 2 = 2(x + 7) − 3.
Solution: We have
−5(x − 4) + 2 = 2(x + 7) − 3
−5x + 20 + 2 = 2x + 14 − 3
−5x + 22 = 2x + 11
−5(x − 4) + 2 = 2(x + 7) − 3
−5x + 22 − 2x = 2x + 11 − 2x
−5x + 20 + 2 = 2x + 14 − 3
−7x + 22 = 11
or, in short,
−7x + 22 − 22 = 11 − 22
−7x = −11
−7x = −11
x=
−11
−7x
=
−7
−7
x=
−5x − 2x = 14 − 3 − 20 − 2
11
7
1
11
7
3. Solve the equation
y+7
y+5 y−2
−
=
+ 1.
2
4
3
Solution: We have
y+7
y+5 y−2
−
=
+1
2
4
3
y+5 y−2
y+7
12 ·
= 12 ·
−
+1
2
4
3
12 ·
y−2
y+7
y+5
− 12 ·
= 12 ·
+ 12 · 1
2
4
3
6(y + 5) − 3(y − 2) = 4(y + 7) + 12
6y + 30 − 3y + 6 = 4y + 28 + 12
3y + 36 = 4y + 40
3y + 36 − 4y = 4y + 40 − 4y
−y + 36 = 40
−y + 36 − 36 = 40 − 36
−y = 4
−y(−1) = 4(−1)
y = −4
In short,
y+5 y−2
y+7
−
=
+1
2
4
3
y+5 y−2
y+7
12 ·
= 12 ·
−
+1
2
4
3
6(y + 5) − 3(y − 2) = 4(y + 7) + 12
6y + 30 − 3y + 6 = 4y + 28 + 12
30 + 6 − 28 − 12 = 4y − 6y + 3y
y = −4
7. Solve for M the equation F = G
Solution: We have
Gm
M
F =
r2
The solution is M =
=⇒
mM
.
r2
r2
Gm
F =
r2 F
.
Gm
2
r2
Gm
Gm
r2
M
=⇒
r2 F
=M
Gm
8. The surface area A of the closed rectangular box can be calculated from the length l, the
width w, and the height h according to the formula
A = 2lw + 2wh + 2lh
Solve for w in terms of the other variables in this equation.
Solution: We have
A = 2lw + 2wh + 2lh
A = (2l + 2h)w + 2lh
A − 2lh = (2l + 2h)w
A − 2lh
=w
2l + 2h
The solution is w =
A − 2lh
.
2l + 2h
Linear Inequalities
An inequality is linear if each term is constant or a multiple of the variable.
EXAMPLE: Solve the inequality 3x < 9x + 4 and sketch the solution set.
Solution: We have
3x < 9x + 4
3x − 9x < 9x + 4 − 9x
−6x < 4
−6x
4
>
−6
−6
x>−
2
3
2
The solution set consists of all numbers greater than − . In other words the solution of the
3
2
inequality is the interval − , ∞ .
3
EXAMPLE: Solve the inequalities 4 ≤ 3x − 2 < 13 and sketch the solution set.
3
EXAMPLE: Solve the inequalities 4 ≤ 3x − 2 < 13 and sketch the solution set.
Solution: We have
4 ≤ 3x − 2 < 13
4 + 2 ≤ 3x − 2 + 2 < 13 + 2
6 ≤ 3x < 15
6
3x
15
≤
<
3
3
3
2≤x<5
Therefore, the solution set is [2, 5).
EXAMPLE: Solve the inequalities −4 < 5 − 3x ≤ 17 and sketch the solution set.
Solution: We have
−4 < 5 − 3x ≤ 17
−4 − 5 < 5 − 3x − 5 ≤ 17 − 5
−9 < −3x ≤ 12
−9
−3x
12
>
≥
−3
−3
−3
3 > x ≥ −4
−4 ≤ x < 3
Therefore, the solution set is [−4, 3).
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