# 2.7 Linear inequalities and absolute value inequalities  (Page 2/11)

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## Using interval notation to express all real numbers less than or equal to a Or greater than or equal to b

Write the interval expressing all real numbers less than or equal to $\text{\hspace{0.17em}}-1\text{\hspace{0.17em}}$ or greater than or equal to $\text{\hspace{0.17em}}1.$

We have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at $\text{\hspace{0.17em}}-\infty \text{\hspace{0.17em}}$ and ends at $\text{\hspace{0.17em}}-1,$ which is written as $\text{\hspace{0.17em}}\left(-\infty ,-1\right].$

The second interval must show all real numbers greater than or equal to $\text{\hspace{0.17em}}1,$ which is written as $\text{\hspace{0.17em}}\left[1,\infty \right).\text{\hspace{0.17em}}$ However, we want to combine these two sets. We accomplish this by inserting the union symbol, $\cup ,$ between the two intervals.

$\left(-\infty ,-1\right]\cup \left[1,\infty \right)$

Express all real numbers less than $\text{\hspace{0.17em}}-2\text{\hspace{0.17em}}$ or greater than or equal to 3 in interval notation.

$\left(-\infty ,-2\right)\cup \left[3,\infty \right)$

## Using the properties of inequalities

When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the addition property and the multiplication property to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.

## Properties of inequalities

These properties also apply to $\text{\hspace{0.17em}}a\le b,$ $a>b,$ and $\text{\hspace{0.17em}}a\ge b.$

Illustrate the addition property for inequalities by solving each of the following:

• (a) $x-15<4$
• (b) $6\ge x-1$
• (c) $x+7>9$

The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.

Solve: $\text{\hspace{0.17em}}3x-2<1.$

$x<1$

## Demonstrating the multiplication property

Illustrate the multiplication property for inequalities by solving each of the following:

1. $3x<6$
2. $-2x-1\ge 5$
3. $5-x>10$

1. $\begin{array}{l}\phantom{\rule{1.5em}{0ex}}3x<6\hfill \\ \frac{1}{3}\left(3x\right)<\left(6\right)\frac{1}{3}\hfill \\ \phantom{\rule{2em}{0ex}}x<2\hfill \end{array}$

Solve: $\text{\hspace{0.17em}}4x+7\ge 2x-3.$

$x\ge -5$

## Solving inequalities in one variable algebraically

As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.

## Solving an inequality algebraically

Solve the inequality: $\text{\hspace{0.17em}}13-7x\ge 10x-4.$

Solving this inequality is similar to solving an equation up until the last step.

The solution set is given by the interval $\text{\hspace{0.17em}}\left(-\infty ,1\right],$ or all real numbers less than and including 1.

Solve the inequality and write the answer using interval notation: $\text{\hspace{0.17em}}-x+4<\frac{1}{2}x+1.$

$\left(2,\infty \right)$

## Solving an inequality with fractions

Solve the following inequality and write the answer in interval notation: $\text{\hspace{0.17em}}-\frac{3}{4}x\ge -\frac{5}{8}+\frac{2}{3}x.$

We begin solving in the same way we do when solving an equation.

The solution set is the interval $\text{\hspace{0.17em}}\left(-\infty ,\frac{15}{34}\right].$

show that the set of all natural number form semi group under the composition of addition
what is the meaning
Dominic
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_3_2_1
felecia
⅗ ⅔½
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_½+⅔-¾
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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
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Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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what is a solution set?
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find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
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16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
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Opoku
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4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
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x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
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-1
Shedrak

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