5x−19≤1 OR −4x+3<−6
5x−19≤1
Add 19 to both sides, then divide by 5.
−4x+3<−6
Subtract 3 from both sides, then divide both sides by -4. Remember that negative divided by/multiplied by a negative = a positive.
5x−19≤1 OR −4x+3<−6
5x ≤ 20 OR -4x < -9
x ≤ 4 OR x > 9/4
Draw a sketch on a number line and you will see that
the solution is set of all numbers, since OR implies you want
the Union of the two sets.
Let's solve these two inequalities step-by-step:
Inequality 1: 5x - 19 ≤ 1
Step 1: Add 19 to both sides of the inequality to isolate the variable:
5x - 19 + 19 ≤ 1 + 19
5x ≤ 20
Step 2: Divide both sides of the inequality by 5 to solve for x:
5x/5 ≤ 20/5
x ≤ 4
Inequality 2: -4x + 3 < -6
Step 1: Subtract 3 from both sides of the inequality to isolate the variable:
-4x + 3 - 3 < -6 - 3
-4x < -9
Step 2: Divide both sides of the inequality by -4. Since we are dividing by a negative number, we need to reverse the inequality symbol:
-4x/-4 > -9/-4
x > 9/4 or x > 2.25
Therefore, the solution to the system of inequalities is x ≤ 4 or x > 2.25.
To solve this compound inequality, we will solve each individual inequality separately and then combine the solutions.
1) 5x - 19 ≤ 1:
Add 19 to both sides of the inequality:
5x - 19 + 19 ≤ 1 + 19
5x ≤ 20
Divide both sides of the inequality by 5 (since we want to isolate x):
5x/5 ≤ 20/5
x ≤ 4
2) -4x + 3 < -6:
Subtract 3 from both sides of the inequality:
-4x + 3 - 3 < -6 - 3
-4x < -9
Divide both sides of the inequality by -4 (remember to reverse the inequality sign when dividing by a negative number):
-4x / -4 > -9 / -4
x > 9/4
Now, combining the solutions:
The first inequality (5x - 19 ≤ 1) gives x ≤ 4, and the second inequality (-4x + 3 < -6) gives x > 9/4.
The combined solution is x ≤ 4 and x > 9/4.
Therefore, the solution for the compound inequality 5x - 19 ≤ 1 OR -4x + 3 < -6 is x ≤ 4, x > 9/4.