Solve the inequality //(2x-1)(x+3)/-4/< 1
where /x/ is the modulus of x.
To solve the inequality in terms of the modulus of x, let's first understand what it means for the inequality to hold.
The modulus of x, denoted as |x|, represents the absolute value of x, which is always a non-negative number. In equation form, |x| = x when x ≥ 0, and |x| = -x when x < 0.
Now let's solve the inequality //(2x-1)(x+3)/-4/< 1 step by step:
Step 1: Determine the critical points.
To find the values of x for which the inequality may change signs, set the expression inside the absolute value symbols equal to zero:
(2x-1)(x+3)/-4 = 0
Solving this equation, we find two critical points:
2x - 1 = 0 --> x = 1/2
x + 3 = 0 --> x = -3
So the critical points are x = 1/2 and x = -3.
Step 2: Test intervals.
To test which intervals of x satisfy the inequality, let's choose test points from each interval. Convenient test points are the critical points found in Step 1 and an additional point outside the intervals.
Let's test three intervals: x < -3, -3 < x < 1/2, and x > 1/2.
For the interval x < -3, we can choose x = -4 as our test point.
(2(-4)-1)(-4+3)/-4 = (-9)(-1)/-4 = 9/4
Since 9/4 > 1, the inequality does not hold for this interval.
For the interval -3 < x < 1/2, we can choose x = 0 as our test point.
(2(0)-1)(0+3)/-4 = (-1)(3)/-4 = 3/4
Since 3/4 < 1, the inequality holds for this interval.
For the interval x > 1/2, we can choose x = 1 as our test point.
(2(1)-1)(1+3)/-4 = (3)(4)/-4 = -3
Since -3 < 1, the inequality holds for this interval.
Step 3: Indicate the solution.
Based on our analysis of the test intervals, we conclude that the inequality //(2x-1)(x+3)/-4/< 1 holds for -3 < x < 1/2.