solve each inequality. Graph the solution.
4 < 3x - 5 ¡Ü 7
5 - x ¡Ý 9 or 3x - 4 > 8
I do not comprehend your question, you have strange symbols here
Solve each inequality. Graph the solution
5 - x ¡Ý 9 or 3x - 4 > 8
I still see ¡Ý
is that supposed to be ≤ ?
To solve each inequality and graph the solution, we will go step by step.
For the first inequality, we have:
4 < 3x - 5 ≤ 7
Step 1: Isolate the term with the variable.
To isolate the variable term, we add 5 to all parts of the inequality:
4 + 5 < 3x - 5 + 5 ≤ 7 + 5
This simplifies to:
9 < 3x ≤ 12
Step 2: Divide by the coefficient of the variable.
To solve for x, divide all parts of the inequality by 3:
9/3 < 3x/3 ≤ 12/3
This simplifies to:
3 < x ≤ 4
Now, let's graph the solution on a number line.
On a number line, we mark a point at 3, but we do not include that point because our inequality is strict inequality (4 < 3x - 5). Then, we draw a line from this point to the right including all values up to 4. We can represent this graphically as:
----------|---o----o-------------
-∞ 3 4 +∞
The open circle at 3 indicates that it is not included in the solution, while the closed circle at 4 indicates that it is included in the solution.
Moving on to the second inequality:
5 - x ≥ 9 or 3x - 4 > 8
Step 1: Simplify each inequality separately.
For the first inequality: 5 - x ≥ 9
Subtract 5 from both sides to isolate x:
5 - 5 - x ≥ 9 - 5
This simplifies to:
-x ≥ 4
Divide by -1, but remember to flip the inequality sign since we are dividing by a negative number:
-x/(-1) ≤ 4/(-1)
This gives us:
x ≤ -4
For the second inequality: 3x - 4 > 8
Add 4 to both sides to isolate x:
3x - 4 + 4 > 8 + 4
This simplifies to:
3x > 12
Divide both sides by 3 to solve for x:
3x/3 > 12/3
This gives us:
x > 4
Step 2: Write the combined solution using "or" to join the two inequalities.
The combined solution is:
x ≤ -4 or x > 4
To graph this solution on a number line, we mark an open circle at -4 and an open circle at 4. Then, we draw a line to the left of -4, including all values less than or equal to -4. Similarly, we draw another line to the right of 4, including all values greater than 4. We can represent this graphically as:
----o-------------------------o----
-∞ -4 4 +∞
The open circles at -4 and 4 indicate that these points are not included in the solution.
Therefore, the graph represents the solution to the inequalities 4 < 3x - 5 ≤ 7 and 5 - x ≥ 9 or 3x - 4 > 8.