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Sagot :
To solve the inequality [tex]\(-4(x+7) < 3(x-2)\)[/tex] step-by-step, we need to simplify both sides of the inequality and isolate [tex]\(x\)[/tex].
1. Expand the expressions:
[tex]\[ -4(x + 7) < 3(x - 2) \][/tex]
Expanding both sides:
[tex]\[ -4x - 28 < 3x - 6 \][/tex]
2. Isolate [tex]\(x\)[/tex] on one side of the inequality:
First, add [tex]\(4x\)[/tex] to both sides to eliminate [tex]\(-4x\)[/tex] from the left side:
[tex]\[ -28 < 3x - 6 + 4x \][/tex]
Simplify:
[tex]\[ -28 < 7x - 6 \][/tex]
Next, add 6 to both sides to eliminate [tex]\(-6\)[/tex] from the right side:
[tex]\[ -28 + 6 < 7x \][/tex]
Simplify:
[tex]\[ -22 < 7x \][/tex]
This can also be written as:
[tex]\[ 7x > -22 \][/tex]
3. Final inequality:
Dividing both sides by 7 changes the expression slightly:
[tex]\[ x > -\frac{22}{7} \][/tex]
Thus the equivalent inequality to the given [tex]\(-4(x+7) < 3(x-2)\)[/tex] is [tex]\(7x > -22\)[/tex].
4. Match with the given options:
To match the provided answer choices:
[tex]\[ 7x > -22 \][/tex]
can be written multiplicative inverse:
[tex]\[ -7 x < 22 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{-7 x < 22} \][/tex]
Thus the correct option is:
D. [tex]\(-7 x < 22\)[/tex]
1. Expand the expressions:
[tex]\[ -4(x + 7) < 3(x - 2) \][/tex]
Expanding both sides:
[tex]\[ -4x - 28 < 3x - 6 \][/tex]
2. Isolate [tex]\(x\)[/tex] on one side of the inequality:
First, add [tex]\(4x\)[/tex] to both sides to eliminate [tex]\(-4x\)[/tex] from the left side:
[tex]\[ -28 < 3x - 6 + 4x \][/tex]
Simplify:
[tex]\[ -28 < 7x - 6 \][/tex]
Next, add 6 to both sides to eliminate [tex]\(-6\)[/tex] from the right side:
[tex]\[ -28 + 6 < 7x \][/tex]
Simplify:
[tex]\[ -22 < 7x \][/tex]
This can also be written as:
[tex]\[ 7x > -22 \][/tex]
3. Final inequality:
Dividing both sides by 7 changes the expression slightly:
[tex]\[ x > -\frac{22}{7} \][/tex]
Thus the equivalent inequality to the given [tex]\(-4(x+7) < 3(x-2)\)[/tex] is [tex]\(7x > -22\)[/tex].
4. Match with the given options:
To match the provided answer choices:
[tex]\[ 7x > -22 \][/tex]
can be written multiplicative inverse:
[tex]\[ -7 x < 22 \][/tex]
So, the correct answer is:
[tex]\[ \boxed{-7 x < 22} \][/tex]
Thus the correct option is:
D. [tex]\(-7 x < 22\)[/tex]
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