Directions:Solve and graph the inequalities using words.
Problems:
4.-3x>-6 or x-5<=2
5.x+2>2x-1 or -6>-2x-2
To solve and graph an inequality using words, we need to break it down step by step. Let's start with problem 4.
4. -3x > -6 or x - 5 ≤ 2
Step 1: Solve the first inequality: -3x > -6
To solve this inequality, we'll start by isolating the variable x. First, divide both sides of the inequality by -3. Remember, when dividing or multiplying by a negative number, we need to flip the inequality sign.
-3x / -3 < -6 / -3
x < 2
So the solution for -3x > -6 is x < 2.
Step 2: Solve the second inequality: x - 5 ≤ 2
For this inequality, we'll again isolate the variable x. First, add 5 to both sides of the inequality.
x - 5 + 5 ≤ 2 + 5
x ≤ 7
So the solution for x - 5 ≤ 2 is x ≤ 7.
Step 3: Combine the solutions from both inequalities.
Now that we have the solutions for both inequalities, we need to find the common solution by taking the intersection. In this case, the common solution is x ≤ 2 and x ≤ 7.
To graph this solution on a number line, draw a line with an open circle at 2 (because x is less than, not less than or equal to), and a closed circle at 7 (because x is less than or equal to). Shade the region between the two points to represent the common solution.
Moving on to problem 5.
5. x + 2 > 2x - 1 or -6 > -2x - 2
Step 1: Solve the first inequality: x + 2 > 2x - 1
To solve this inequality, we'll again start by isolating the variable x. First, subtract x from both sides.
x + 2 - x > 2x - x - 1
2 > x - 1
Step 2: Solve the second inequality: -6 > -2x - 2
For this inequality, we'll isolate the variable x by adding 2x to both sides and subtracting 6 from both sides.
-6 + 6 > -2x - 2 + 2x
0 > x - 2
Step 3: Combine the solutions from both inequalities.
Now, we have x - 2 > 0 and 2 > x - 1.
To find the common solution, we need to look for the intersection of these two inequalities. In this case, the common solution is x > 2.
To graph this solution on a number line, draw an open circle at 2 (because x is greater than, not greater than or equal to). Then shade the region to the right of 2, representing the solutions where x is greater than 2.
Remember, to represent an open circle on the number line, use an empty circle, and for a closed circle, use a filled-in circle. Also, make sure to label the number line with the appropriate values of x.