Solve and graph the solution to the compound inequality.
−4x ≥ 36 or x − 7 ≥ −3
(1 point)
To solve the compound inequality, we will solve each inequality separately and then combine the solutions.
First, let's solve the inequality -4x ≥ 36:
Dividing both sides by -4 (remembering to reverse the inequality when dividing by a negative number), we have:
x ≤ -9
Now let's solve the inequality x - 7 ≥ -3:
Adding 7 to both sides, we have:
x ≥ 4
Now, let's combine the solutions:
The solution to the compound inequality is x ≤ -9 or x ≥ 4.
To graph the solution on a number line, we draw a number line and mark -9 and 4 with open circles (since the inequalities are "greater than or equal to" and "less than or equal to"). Then, we shade the area to the left of -9 and the area to the right of 4, including the two points.
Here is the graph:
<=========o====[<------------------]====o===>
-9 4
To solve a compound inequality, break it down into two separate inequalities and find the solutions to each one. Then combine the solutions to get the final solution.
Starting with the first inequality:
-4x ≥ 36
To solve this, divide both sides of the inequality by -4. Since we are dividing by a negative number, we need to reverse the inequality sign.
Dividing both sides by -4, we get:
x ≤ -9
Now moving to the second inequality:
x - 7 ≥ -3
To solve this, add 7 to both sides of the inequality:
x ≥ 4
Now we have the solutions to each inequality:
x ≤ -9
x ≥ 4
To find the final solution, we need to find the intersection of these two solutions. In other words, find the values of x that satisfy both inequalities.
The graph of x ≤ -9 would include all values less than or equal to -9, represented by a closed circle on -9 and shading to the left.
The graph of x ≥ 4 would include all values greater than or equal to 4, represented by a closed circle on 4 and shading to the right.
To represent the intersection of these two graphs, we need to shade the region that is common to both graphs. Since there is no overlap between the two graphs, the final solution is an empty set.
The graph of the compound inequality would be an empty set or the symbol Ø, indicating that there are no values that satisfy both inequalities.
To solve the compound inequality, we need to solve each individual inequality separately and then combine the solutions.
Starting with the first inequality:
-4x ≥ 36
Step 1: Divide both sides of the inequality by -4, remembering that when we multiply or divide both sides of an inequality by a negative number, we need to reverse the inequality sign.
x ≤ -9
Now, moving on to the second inequality:
x - 7 ≥ -3
Step 2: Add 7 to both sides of the inequality.
x ≥ 4
Combining the two solutions we have:
x ≤ -9 or x ≥ 4
Now, let's graph the solution on the number line:
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🠔 🠖
-∞ -9 4 ∞
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To graph the solution, we represent the values x ≤ -9 with a closed circle on -9 and an arrow pointing to the left, indicating that all values to the left of -9 are included. Similarly, we represent the values x ≥ 4 with a closed circle on 4 and an arrow pointing to the right, indicating that all values to the right of 4 are included.
The shaded region between the two arrows represents the solution to the compound inequality.