Math / Algebra

QuestionSolve and graph the following inequality. $1-4 x \geq 25 \text { and } 7 x-4>10$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is $\{x|x\rangle$ $\square$ 3 B. The solution set is $\{x \mid$ $\square$ $\leq x<$ $\square$ 3 C. The solution set is $\{x \mid x \geq$ $\square$ 3 D. The solution set is $\{x \mid$ $\square$ $<x \leq$ $\square$ E. The solution set is $\varnothing$ .
Graph the solution. Choose the correct answer below. A. B. c. D. E. O F.

Studdy Solution

STEP 1

1. We have two inequalities to solve: $1 - 4x \geq 25$ and $7x - 4 > 10$ .
2. We need to find the solution set that satisfies both inequalities.
3. We will graph the solution set on a number line.

STEP 2

1. Solve the first inequality.
2. Solve the second inequality.
3. Find the intersection of the solution sets.
4. Graph the solution set.

STEP 3

Solve the first inequality: $1 - 4x \geq 25$ .
Subtract 1 from both sides:
$-4x \geq 24$
Divide both sides by -4, remembering to reverse the inequality sign:
$x \leq -6$

STEP 4

Solve the second inequality: $7x - 4 > 10$ .
Add 4 to both sides:
$7x > 14$
Divide both sides by 7:
$x > 2$

STEP 5

Find the intersection of the solution sets from the two inequalities.
The first inequality gives $x \leq -6$ . The second inequality gives $x > 2$ .
Since there is no overlap between $x \leq -6$ and $x > 2$ , the solution set is empty.

STEP 6

Graph the solution set.
Since the solution set is empty, there is nothing to graph on the number line.
The correct choice is:
E. The solution set is $\varnothing$ .

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Studdy Solution

STEP 1

1. We have two inequalities to solve: $1 - 4x \geq 25$ and $7x - 4 > 10$ .
2. We need to find the solution set that satisfies both inequalities.
3. We will graph the solution set on a number line.

STEP 2

1. Solve the first inequality.
2. Solve the second inequality.
3. Find the intersection of the solution sets.
4. Graph the solution set.

STEP 3

Solve the first inequality: $1 - 4x \geq 25$ .
Subtract 1 from both sides:
$-4x \geq 24$
Divide both sides by -4, remembering to reverse the inequality sign:
$x \leq -6$

STEP 4

Solve the second inequality: $7x - 4 > 10$ .
Add 4 to both sides:
$7x > 14$
Divide both sides by 7:
$x > 2$

STEP 5

Find the intersection of the solution sets from the two inequalities.
The first inequality gives $x \leq -6$ . The second inequality gives $x > 2$ .
Since there is no overlap between $x \leq -6$ and $x > 2$ , the solution set is empty.

STEP 6

Graph the solution set.
Since the solution set is empty, there is nothing to graph on the number line.
The correct choice is:
E. The solution set is $\varnothing$ .

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Studdy Solution

STEP 1

1. We have two inequalities to solve: 1−4x≥251 - 4x \geq 251−4x≥25 and 7x−4>107x - 4 > 107x−4>10.2. We need to find the solution set that satisfies both inequalities.3. We will graph the solution set on a number line.

STEP 2

1. Solve the first inequality.2. Solve the second inequality.3. Find the intersection of the solution sets.4. Graph the solution set.

STEP 3

Solve the first inequality: 1−4x≥251 - 4x \geq 251−4x≥25.Subtract 1 from both sides:−4x≥24 -4x \geq 24 −4x≥24Divide both sides by -4, remembering to reverse the inequality sign:x≤−6 x \leq -6 x≤−6

STEP 4

Solve the second inequality: 7x−4>107x - 4 > 107x−4>10.Add 4 to both sides:7x>14 7x > 14 7x>14Divide both sides by 7:x>2 x > 2 x>2

STEP 5

Find the intersection of the solution sets from the two inequalities.The first inequality gives x≤−6 x \leq -6 x≤−6. The second inequality gives x>2 x > 2 x>2.Since there is no overlap between x≤−6 x \leq -6 x≤−6 and x>2 x > 2 x>2, the solution set is empty.

STEP 6

Graph the solution set.Since the solution set is empty, there is nothing to graph on the number line. The correct choice is:E. The solution set is ∅ \varnothing ∅.

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1. We have two inequalities to solve: $1 - 4x \geq 25$ and $7x - 4 > 10$ .
2. We need to find the solution set that satisfies both inequalities.
3. We will graph the solution set on a number line.

1. Solve the first inequality.
2. Solve the second inequality.
3. Find the intersection of the solution sets.
4. Graph the solution set.

Solve the first inequality: $1 - 4x \geq 25$ .
Subtract 1 from both sides:
$-4x \geq 24$
Divide both sides by -4, remembering to reverse the inequality sign:
$x \leq -6$

Solve the second inequality: $7x - 4 > 10$ .
Add 4 to both sides:
$7x > 14$
Divide both sides by 7:
$x > 2$

Find the intersection of the solution sets from the two inequalities.
The first inequality gives $x \leq -6$ . The second inequality gives $x > 2$ .
Since there is no overlap between $x \leq -6$ and $x > 2$ , the solution set is empty.

Graph the solution set.
Since the solution set is empty, there is nothing to graph on the number line.
The correct choice is:
E. The solution set is $\varnothing$ .