Compound Inequalities 2

Topics

Pre - Algebra

Inequalities

Compound Inequalities

Compound Inequalities 2

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Question 1 of 4

1. Question
Choose the number line that represents the inequality

$x \geq 1$ $x≥1$ OR $x$ $x$ $<$ $<$ $- 2$ $-2$
- 1.
- 2.
- 3.
- 4.
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A compound inequality consists of two inequalities joined together by AND or OR.

Plot $x \geq$ $x≥$ $1$ $1$ first.

The given inequality has a greater than or equal to ( $\geq$ $≥$ ) sign.

Hence, place a solid circle above $1$ $1$ and attach an arrow pointing to the right to represent all values greater than $1$ $1$ .

Next, plot $x$ $x$ $<$ $<$ $- 2$ $-2$ .

The given inequality has a less than ( $<$ $<$ ) sign.

Hence, place an empty circle above $- 2$ $-2$ and attach an arrow pointing to the left to represent all values less than $- 2$ $-2$ .

Finally, combine the two number lines.
Question 2 of 4

2. Question
Find the inequality that represents all possible $x$ $x$ values

$1$ $1$ $<$ $<$ $x + 3 \leq 9$ $x+3≤9$
- 1. $- 2$ $-2$ $>$ $>$ $x \geq 6$ $x≥6$
- 2. $2$ $2$ $>$ $>$ $x \geq - 6$ $x≥-6$
- 3. $2$ $2$ $<$ $<$ $x \leq - 6$ $x≤-6$
- 4. $- 2$ $-2$ $<$ $<$ $x \leq 6$ $x≤6$
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A compound inequality consists of two inequalities joined together by AND or OR.

First, make sure that $x$ $x$ is left alone in the middle

$1$ $1$ $<$ $<$ $x + 3$ $x+3$ $\leq$ $≤$ $9$ $9$

$1$ $1$ $- 3$ $-3$ $<$ $<$ $x + 3$ $x+3$ $- 3$ $-3$ $\leq$ $≤$ $9$ $9$ $- 3$ $-3$ Subtract $3$ $3$ from all sections

$- 2$ $-2$ $<$ $<$ $x$ $x$ $\leq$ $≤$ $6$ $6$

$- 2$ $-2$ $<$ $<$ $x \leq 6$ $x≤6$

Question 3 of 4

Find the inequality that represents all possible

x

$x$ values

- 6 \leq - x - 5 \leq 2

$-6≤-x-5≤2$

1. $7 \leq x \leq - 1$ $7≤x≤-1$
2. $- 7 \leq x \leq 1$ $-7≤x≤1$
3. $- 7 \geq x \geq 1$ $-7≥x≥1$
4. $7 \geq x \geq - 1$ $7≥x≥-1$

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A compound inequality consists of two inequalities joined together by AND or OR.

First, make sure that

x

$x$ is left alone in the middle

$- 6 \leq - x - 5$ $-6≤-x-5$	$\leq$ $≤$	$2$ $2$
$- 6$ $-6$ $+ 5$ $+5$ $\leq - x - 5$ $≤-x-5$ $+ 5$ $+5$	$\leq$ $≤$	$2$ $2$ $+ 5$ $+5$	Add $5$ $5$ to all sections
$- 1$ $-1$ $\times (- 1)$ $times(-1)$ $\leq - x$ $≤-x$ $\times (- 1)$ $times(-1)$	$\leq$ $≤$	$7$ $7$ $\times (- 1)$ $times(-1)$	Multiply all sections by $- 1$ $-1$
$1 \geq x$ $1≥x$	$\geq$ $≥$	$- 7$ $-7$	Flip the inequality
$- 7 \leq x$ $-7≤x$	$\leq$ $≤$	$1$ $1$	Place the lower value to the left

- 7 \leq x \leq 1

$-7≤x≤1$

Question 4 of 4

Find the inequality that represents all possible

x

$x$ values

10

$10$

>

$>$

2 - 4 x

$2-4x$ AND

7 - 4 x

$7-4x$

>

$>$

- 5

$-5$

1. $2$ $2$ $>$ $>$ $x$ $x$ $>$ $>$ $- 3$ $-3$
2. $- 2$ $-2$ $<$ $<$ $x$ $x$ $<$ $<$ $3$ $3$
3. $2$ $2$ $>$ $>$ $x$ $x$ $>$ $>$ $- 3$ $-3$
4. $- 2$ $-2$ $<$ $<$ $x$ $x$ $<$ $<$ $3$ $3$

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A compound inequality consists of two inequalities joined together by AND or OR.

Solve the two inequalities seperately.

Solve for

10

$10$

>

$>$

2 - 4 x

$2-4x$ first.

$10$ $10$	$>$ $>$	$2 - 4 x$ $2-4x$
$10$ $10$ $- 2$ $-2$	$>$ $>$	$2 - 4 x$ $2-4x$ $- 2$ $-2$	Subtract $2$ $2$ from both sides
$8$ $8$ $\div (- 4)$ $divide(-4)$	$>$ $>$	$- 4 x$ $-4x$ $\div (- 4)$ $divide(-4)$	Divide both sides by $- 4$ $-4$
$- 2$ $-2$	$<$ $<$	$x$ $x$	Flip the inequality

Next, solve for

7 - 4 x

$7-4x$

>

$>$

- 5

$-5$ .

$7 - 4 x$ $7-4x$	$>$ $>$	$- 5$ $-5$
$7 - 4 x$ $7-4x$ $- 7$ $-7$	$>$ $>$	$- 5$ $-5$ $- 7$ $-7$	Subtract $7$ $7$ from both sides
$- 4 x$ $-4x$ $\div (- 4)$ $divide(-4)$	$>$ $>$	$- 12$ $-12$ $\div (- 4)$ $divide(-4)$	Divide both sides by $- 4$ $-4$
$x$ $x$	$<$ $<$	$3$ $3$	Flip the inequality

The two inequalities can be rewritten as:

- 2

$-2$

<

$<$

x

$x$

<

$<$

3

$3$

- 2

$-2$

<

$<$

x

$x$

<

$<$

3

$3$

Compound Inequalities 2

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Quiz summary

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1. Question

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2. Question

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3. Question

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4. Question

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Quizzes

$1$ $1$ $<$ $<$ $x + 3$ $x+3$	$\leq$ $≤$	$9$ $9$
$1$ $1$ $- 3$ $-3$ $<$ $<$ $x + 3$ $x+3$ $- 3$ $-3$	$\leq$ $≤$	$9$ $9$ $- 3$ $-3$	Subtract $3$ $3$ from all sections
$- 2$ $-2$ $<$ $<$ $x$ $x$	$\leq$ $≤$	$6$ $6$