Solve a system of linear inequalities - High school Algebra I Exercises

Solve the following system of linear inequalities.

\begin{cases} 2x+3 \lt 1 \cr \cr 4-x≥3 \end{cases}

x \lt -2

x \lt -5

x \lt -1

x \lt 3

In order to solve a system of inequalities, we proceed as follows:

Solve each inequality independently.
Find numbers that satisfy all inequalities.

Step 1

Solve the first inequality

2x+3 \lt 1

Subtract 3 from both sides:

2x+3-3 \lt 1-3

2x \lt -2

Divide both sides by 2:

\dfrac{2x}{2} \lt \dfrac{-2}{2}

x \lt -1

Step 2

Solve the second inequality

4-x≥3

Subtract 4 from both sides:

4-x-4≥3-4

-x \geq -1

Divide by -1:

x \leq 1

Step 3

Find solution set of the system

Find all numbers that satisfy both:

x \lt -1
x \leq 1

That is to say:

x \lt -1

The solution set of the system is \left( -\infty,-1 \right).

\begin{cases} 2x+4 \lt 10 \cr \cr 4-2x≥6 \end{cases}

-7\leq x \lt 4

-9\leq x \lt -4

x \leq -1

-6\leq x

In order to solve a system of inequalities, we proceed as follows:

Solve each inequality independently.
Find numbers that satisfy all inequalities.

Step 1

Solve the first inequality

2x+4 \lt 10

Subtract 4 from both sides:

2x+4-4 \lt 10-4

2x \lt 6

Divide both sides by 2:

\dfrac{2x}{2} \lt \dfrac{6}{2}

x \lt 3

Step 2

Solve the second inequality

4-2x \geq 6

Subtract 4 from both sides:

4-2x -4\geq 6-4

-2x \geq 2

Divide both sides by -2:

\dfrac{-2x}{-2} \leq \dfrac{2}{-2}

x \leq -1

Step 3

Find solution set of the system

Find all numbers that satisfy both:

x \lt 3
x \leq -1

That is to say:

x \leq -1

The solution set of the system is \left( -\infty,-1 \right].

\begin{cases} x+4 \lt 8 \cr \cr 2-x≥5 \end{cases}

x \leq -3

-5\leq x

-5\leq x \lt 4

-2\leq x \lt 5

In order to solve a system of inequalities, we proceed as follows:

Solve each inequality independently.
Find numbers that satisfy all inequalities.

Step 1

Solve the first inequality

x+4 \lt 8

Subtract 4 from both sides:

x+4-4 \lt 8-4

x \lt 4

Step 2

Solve the second inequality

2-x≥5

Subtract 2 from both sides:

2-x-2≥5-2

-x \geq 3

Divide by -1:

x \leq -3

Step 3

Find solution set of the system

Find all numbers that satisfy both:

x \lt 4
x \leq -3

That is to say:

x \leq -3

The solution set of the system is \left( -\infty,-3 \right].

\begin{cases} 2x \lt 12 \cr \cr -x≥7 \end{cases}

x \leq -7

-5\leq x \lt 4

x \lt 14

-9\leq x \lt 14

In order to solve a system of inequalities, we proceed as follows:

Solve each inequality independently.
Find numbers that satisfy all inequalities.

Step 1

Solve the first inequality

2x \lt 12

Divide both sides by 2:

\dfrac{2x}{2} \lt \dfrac{12}{2}

x \lt 6

Step 2

Solve the second inequality

-x≥7

Divide by -1:

x \leq -7

Step 3

Find solution set of the system

Find all numbers that satisfy both:

x \lt 6
x \leq -7

That is to say

x \leq -7

The solution of the system is \left( -\infty,-7 \right].

\begin{cases} 4x+8 \lt 24 \cr \cr 2-2x≥14 \end{cases}

-1\leq x \lt 3

-5\leq x

-5\leq x \lt 4

x \leq -6

In order to solve a system of inequalities, we proceed as follows:

Solve each inequality independently.
Find numbers that satisfy all inequalities.

Step 1

Solve the first inequality

4x+8 \lt 24

Subtract 8 from both sides:

4x+8-8 \lt 24-8

4x \lt 16

Divide both sides by 4:

\dfrac{4x}{4} \lt \dfrac{16}{4}

x \lt 4

Step 2

Solve the second inequality

2-2x≥14

Subtract 2 from both sides:

2-2x-2≥14-2

-2x \geq 12

Divide by -2:

x \leq -6

Step 3

Find solution set of the system

Find all numbers that satisfy both:

x \lt 4
x \leq -6

That is to say:

x \leq -6

The solution set of the system is \left(- \infty,-6 \right].

\begin{cases} 3x+6 \lt 18 \cr \cr 4-4x\leq24 \end{cases}

x \lt 3

-5\leq x \lt 4

-3\leq x

-1\leq x \lt 7

In order to solve a system of inequalities, we proceed as follows:

Solve each inequality independently.
Find numbers that satisfy all inequalities.

Step 1

Solve the first inequality

3x+6 \lt 18

Subtract 6 from both sides:

3x+6-6 \lt 18-6

3x \lt 12

Divide both sides by 3:

\dfrac{3x}{3} \lt \dfrac{12}{3}

x \lt 4

Step 2

Solve the second inequality

4-4x\leq 24

Subtract 4 from both sides:

4-4x-4\leq24-4

-4x \leq 20

Divide by -4:

x \geq -5

Step 3

Find solution set of the system

Find all numbers that satisfy both:

x \lt 4
x \geq -5

That is to say:

-5\leq x \lt 4

The solution of the system is \left[ -5{,}4 \right).

\begin{cases} -3x+3 \gt 18 \cr \cr 10+4x≥12 \end{cases}

x \geq \dfrac{1}{2}

x \leq \dfrac{-1}{2}

x \geq \dfrac{5}{3}

x \geq \dfrac{12}{5}

In order to solve a system of inequalities, we proceed as follows:

Solve each inequality independently.
Find numbers that satisfy all inequalities.

Step 1

Solve the first inequality

-3x+3 \gt 18

Subtract 3 from both sides:

-3x+3-3 \gt 18-3

-3x \gt 15

Divide both sides by -3:

\dfrac{-3x}{-3} \gt \dfrac{15}{-3}

x \gt -5

Step 2

Solve the second inequality

10+4x≥12

Subtract 10 from both sides:

10+4x-10≥12-10

4x \geq 2

Divide by 4:

x \geq \dfrac{1}{2}

Step 3

Find solution set of the system

Find all numbers that satisfy both:

x \gt -5
x \geq \dfrac{1}{2}

That is to say:

x \geq \dfrac{1}{2}

The solution of the system is \left[ \dfrac{1}{2},\infty \right).