Math / Algebra

QuestionGroupe: 303
Numéro: $f$ a) $x+9>15$
Intervalle: $x+9>15$ $-9-9$ b) $4 x>12$ $x>6 \quad x \in] 6,+\infty[$ $3 x+9<4 x+7$ $2 x+5 \leq-11 x+31$ $-2(x+6)<-18$ $2(x+3) \leq 4(x-1)$ i) $1,25 x+2(x-0,25)>-0,75 x$

Studdy Solution

STEP 1

1. We are solving a series of algebraic inequalities.
2. Each inequality needs to be solved for $x$ .
3. The solutions should be expressed in interval notation where applicable.
4. The inequalities involve basic algebraic operations, including addition, subtraction, multiplication, and division.

STEP 2

1. Solve the inequality $x + 9 > 15$ .
2. Solve the inequality $4x > 12$ .
3. Solve the inequality $3x + 9 < 4x + 7$ .
4. Solve the inequality $2x + 5 \leq -11x + 31$ .
5. Solve the inequality $-2(x + 6) < -18$ .
6. Solve the inequality $2(x + 3) \leq 4(x - 1)$ .
7. Solve the inequality $1.25x + 2(x - 0.25) > -0.75x$ .

STEP 3

Solve the inequality $x + 9 > 15$ .
Subtract 9 from both sides:
$x + 9 - 9 > 15 - 9$
Simplify:
$x > 6$
Express the solution in interval notation:
$x \in (6, +\infty)$

STEP 4

Solve the inequality $4x > 12$ .
Divide both sides by 4:
$\frac{4x}{4} > \frac{12}{4}$
Simplify:
$x > 3$
Express the solution in interval notation:
$x \in (3, +\infty)$

STEP 5

Solve the inequality $3x + 9 < 4x + 7$ .
Subtract $3x$ from both sides:
$9 < x + 7$
Subtract 7 from both sides:
$9 - 7 < x$
Simplify:
$2 < x$
Express the solution in interval notation:
$x \in (2, +\infty)$

STEP 6

Solve the inequality $2x + 5 \leq -11x + 31$ .
Add $11x$ to both sides:
$2x + 11x + 5 \leq 31$
Combine like terms:
$13x + 5 \leq 31$
Subtract 5 from both sides:
$13x \leq 26$
Divide both sides by 13:
$x \leq 2$
Express the solution in interval notation:
$x \in (-\infty, 2]$

STEP 7

Solve the inequality $-2(x + 6) < -18$ .
Distribute the $-2$ :
$-2x - 12 < -18$
Add 12 to both sides:
$-2x < -6$
Divide both sides by $-2$ (remember to reverse the inequality sign):
$x > 3$
Express the solution in interval notation:
$x \in (3, +\infty)$

STEP 8

Solve the inequality $2(x + 3) \leq 4(x - 1)$ .
Distribute on both sides:
$2x + 6 \leq 4x - 4$
Subtract $2x$ from both sides:
$6 \leq 2x - 4$
Add 4 to both sides:
$10 \leq 2x$
Divide both sides by 2:
$5 \leq x$
Express the solution in interval notation:
$x \in [5, +\infty)$

STEP 9

Solve the inequality $1.25x + 2(x - 0.25) > -0.75x$ .
Distribute the 2:
$1.25x + 2x - 0.5 > -0.75x$
Combine like terms:
$3.25x - 0.5 > -0.75x$
Add $0.75x$ to both sides:
$4x - 0.5 > 0$
Add 0.5 to both sides:
$4x > 0.5$
Divide both sides by 4:
$x > 0.125$
Express the solution in interval notation:
$x \in (0.125, +\infty)$

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QuestionGroupe: 303
Numéro: $f$ a) $x+9>15$
Intervalle: $x+9>15$ $-9-9$ b) $4 x>12$ $x>6 \quad x \in] 6,+\infty[$ $3 x+9<4 x+7$ $2 x+5 \leq-11 x+31$ $-2(x+6)<-18$ $2(x+3) \leq 4(x-1)$ i) $1,25 x+2(x-0,25)>-0,75 x$

Studdy Solution

STEP 1

1. We are solving a series of algebraic inequalities.
2. Each inequality needs to be solved for $x$ .
3. The solutions should be expressed in interval notation where applicable.
4. The inequalities involve basic algebraic operations, including addition, subtraction, multiplication, and division.

STEP 2

STEP 3

Solve the inequality $x + 9 > 15$ .
Subtract 9 from both sides:
$x + 9 - 9 > 15 - 9$
Simplify:
$x > 6$
Express the solution in interval notation:
$x \in (6, +\infty)$

STEP 4

Solve the inequality $4x > 12$ .
Divide both sides by 4:
$\frac{4x}{4} > \frac{12}{4}$
Simplify:
$x > 3$
Express the solution in interval notation:
$x \in (3, +\infty)$

STEP 5

Solve the inequality $3x + 9 < 4x + 7$ .
Subtract $3x$ from both sides:
$9 < x + 7$
Subtract 7 from both sides:
$9 - 7 < x$
Simplify:
$2 < x$
Express the solution in interval notation:
$x \in (2, +\infty)$

STEP 6

Solve the inequality $2x + 5 \leq -11x + 31$ .
Add $11x$ to both sides:
$2x + 11x + 5 \leq 31$
Combine like terms:
$13x + 5 \leq 31$
Subtract 5 from both sides:
$13x \leq 26$
Divide both sides by 13:
$x \leq 2$
Express the solution in interval notation:
$x \in (-\infty, 2]$

STEP 7

Solve the inequality $-2(x + 6) < -18$ .
Distribute the $-2$ :
$-2x - 12 < -18$
Add 12 to both sides:
$-2x < -6$
Divide both sides by $-2$ (remember to reverse the inequality sign):
$x > 3$
Express the solution in interval notation:
$x \in (3, +\infty)$

STEP 8

Solve the inequality $2(x + 3) \leq 4(x - 1)$ .
Distribute on both sides:
$2x + 6 \leq 4x - 4$
Subtract $2x$ from both sides:
$6 \leq 2x - 4$
Add 4 to both sides:
$10 \leq 2x$
Divide both sides by 2:
$5 \leq x$
Express the solution in interval notation:
$x \in [5, +\infty)$

STEP 9

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Studdy Solution

STEP 1

1. We are solving a series of algebraic inequalities.2. Each inequality needs to be solved for x x x.3. The solutions should be expressed in interval notation where applicable.4. The inequalities involve basic algebraic operations, including addition, subtraction, multiplication, and division.

STEP 2

STEP 3

Solve the inequality x+9>15 x + 9 > 15 x+9>15.Subtract 9 from both sides:x+9−9>15−9 x + 9 - 9 > 15 - 9 x+9−9>15−9Simplify:x>6 x > 6 x>6Express the solution in interval notation:x∈(6,+∞) x \in (6, +\infty) x∈(6,+∞)

STEP 4

Solve the inequality 4x>12 4x > 12 4x>12.Divide both sides by 4:4x4>124 \frac{4x}{4} > \frac{12}{4} 44x​>412​Simplify:x>3 x > 3 x>3Express the solution in interval notation:x∈(3,+∞) x \in (3, +\infty) x∈(3,+∞)

STEP 5

Solve the inequality 3x+9<4x+7 3x + 9 < 4x + 7 3x+9<4x+7.Subtract 3x 3x 3x from both sides:9<x+7 9 < x + 7 9<x+7Subtract 7 from both sides:9−7<x 9 - 7 < x 9−7<xSimplify:2<x 2 < x 2<xExpress the solution in interval notation:x∈(2,+∞) x \in (2, +\infty) x∈(2,+∞)

STEP 6

STEP 7

STEP 8

STEP 9

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. We are solving a series of algebraic inequalities.
2. Each inequality needs to be solved for $x$ .
3. The solutions should be expressed in interval notation where applicable.
4. The inequalities involve basic algebraic operations, including addition, subtraction, multiplication, and division.

Solve the inequality $x + 9 > 15$ .
Subtract 9 from both sides:
$x + 9 - 9 > 15 - 9$
Simplify:
$x > 6$
Express the solution in interval notation:
$x \in (6, +\infty)$

Solve the inequality $4x > 12$ .
Divide both sides by 4:
$\frac{4x}{4} > \frac{12}{4}$
Simplify:
$x > 3$
Express the solution in interval notation:
$x \in (3, +\infty)$

Solve the inequality $3x + 9 < 4x + 7$ .
Subtract $3x$ from both sides:
$9 < x + 7$
Subtract 7 from both sides:
$9 - 7 < x$
Simplify:
$2 < x$
Express the solution in interval notation:
$x \in (2, +\infty)$